of semi-simplicial types. Roughly speaking, a semi-simplicial type 
consists of a type 
together with, for each 
, a displayed semi-simplicial type over 
. This mimics how simplices can be generated geometrically through repeated cones, and is made possible by the display primitive at the simplicial mode. The discrete part of 
then yields the usual infinite indexed definition of semi-simplicial types, both semantically and syntactically. Thus, dTT enables working with semi-simplicial types in full semantic generality.
1. Introduction
Semi-simplicial types
Homotopy Type Theory (HoTT) Univalent Foundations Program (2013) is a perspective on intensional dependent type theory that regards types as homotopical spaces or
$\infty$
-groupoids. It has proven remarkably successful as a synthetic context in which to do homotopy theory and algebraic topology, and as an internal language for
$(\infty,1)$
-toposes Shulman (Reference Shulman2019). However, an enduring frustration has been its apparent inability to define general homotopy-coherent structures. Some infinite structures can be defined in HoTT, such as globular types and spectra; but others, such as
$A_\infty$
-spaces or
$(\infty,1)$
-categories, have so far resisted all attempts at definition. We know no convincing explanation for why they should be impossible, but the fact that all attempts appear to fail in a similar way suggests the operation of an as-yet-unarticulated principle.
Specifically, stating an ‘infinite coherence’ property generally seems to require an infinite structure within which to assemble the coherences, while defining such a structure itself seems to require infinite coherence, leading to an infinite regress. This is in contrast to the situation in classical homotopy theory where the infinite structures to describe coherence, such as operads and simplicial diagrams, can themselves be defined using strict point-set-level equalities, which are then automatically fully coherent. It is tempting to try to mimic this in homotopy type theory using definitional equalities in place of point-set ones, but this is difficult because definitional equality is not reified in the theory and we have limited tools for forcing it to hold.
One of the more flexible ways to enforce definitional equalities is to use type dependency, moving from a fibred perspective to an indexed one. In the simplest case, this means replacing a function 
with a type family 
. This has a corresponding projection function 
, and we can suppose a point 
with a definitional equality 
by supposing a point 
and letting 
.
Thus, it is natural to try to define infinitely coherent structures that can be expressed in a purely indexed way. The example of this sort which has attracted the most attention is that of semi-simplicial types, because they are well-known within homotopy theory and could be used to encode many, if not all, other infinitely coherent structures. In indexed style, a semi-simplicial type consists of type families 
, and so on, having types that start out as follows:
First, we have a type 
of points. Second, we have for every two points 
, a type 
of lines joining 
and 
. Third, for every three points 
and three lines 
, 
, a type 
of triangles with the given boundary. The pattern continues with tetrahedra, which we may also, more technically, call 
-simplices. In general, 
describes the type of 
-simplices indexed by their boundaries.
Remark 1. The binary subscripts on variables above follow a scheme that we learned from Tim Campion, although related schemas have been rediscovered many times. The 
simplices constituting an 
-simplex and its boundary are labelled by the numbers from 
to 
written in binary, where the number of 
s in a binary number corresponds to the dimension of the simplex, and the binary numbers corresponding to the boundary of some simplex are obtained by replacing one or more of the 
s in its binary number by 
s. (As we will see later, the binary number 
can also be regarded as denoting the unique 
-simplex in the boundary of an 
-simplex in an augmented semi-simplicial set.)
We then list these simplices in the order given by these numbers. This ordering may seem somewhat curious, compared to the more naïve approach of listing all the 
-simplices, then all the 
-simplices, and so on; but it does retain the important property that all the simplices in the boundary of some simplex are listed before it (thus, for instance, it makes the semi-simplex category into an ‘ordered direct category’; see section 4.5.5). We will see later that this ordering is what arises most naturally in (co)inductive constructions of semi-simplicial sets (which was also Campion’s motivation).
In addition to subscripting simplex variables by binary numbers according to this scheme, in this paper we will use different base letters to indicate the dimension of each variable. Thus for instance 
are all 
-simplices, while 
, are all 
-simplices, and 
is a 
-simplex. We will not have much occasion to denote 
-simplices; for 
-simplices we use the Cyrillic letter 
(ze). We may also use different letters from the same alphabets for simplices in different semi-simplicial types, e.g. 
and 
.
One may think of the terms 
and so on as defining the fields of an infinite record type. Terms of this infinite record type 
are known as semi-simplicial types. Thus, the problem is to define a type of semi-simplicial types within homotopy type theory, continuing the above pattern. As a correctness criterion, one would expect that when interpreted in any
$(\infty,1)$
-topos the type 
becomes a classifier of semi-simplicial objects. However, every attempt to internally encode the combinatorics that generate the type of 
, as a function of 
, seems to lead once again to an infinite regress.
In light of this situation, an alternative approach is to formulate more expressive type theories that can solve the problem of infinitely coherent objects. One such proposal is Two-Level Type Theory (2LTT) Annenkov et al. (Reference Annenkov, Capriotti, Kraus and Sattler2023), which introduces an ‘outer level’ of ‘exo-types’ that are not homotopy-invariant. The exo-types admit a strict exo-equality type, essentially reifying definitional equality, which can then be used analogously to classical point-set equality to define infinitely coherent structures. And by the results of Uskuplu (Reference Uskuplu2023), 2LTT can be interpreted in any
$(\infty,1)$
-topos, so its semantics are not significantly less general than ordinary HoTT, and the type 
defined in 2LTT does interpret to the correct classifier. However, although the exo-equality is assumed to satisfy Uniqueness of Identity Proofs, to keep type-checking decidable it cannot satisfy a reflection rule making exo-equalities into definitional equalities. Thus, it can be quite cumbersome to work with in practice.
Another proposal is Simplicial Type Theory (STT) Riehl and Shulman (Reference Riehl and Shulman2017), which changes perspective to view individual types as simplicial spaces, with additional primitives for manipulating the simplicial structure. One can then simply impose conditions on one of these ‘simplicial types’ to make it represent (for instance) an
$(\infty,1)$
-category. This suggests a ‘synthetic’ approach to higher category theory analogous to ordinary HoTT’s synthetic approach to homotopy theory, which is potentially quite powerful; and the results of Riehl and Shulman (Reference Riehl and Shulman2017); Weinberger (Reference Weinberger2022) imply that it can be interpreted in the category of simplicial objects in any
$(\infty,1)$
-topos. However, the strength of the synthetic approach is also its weakness: because simplicial types are postulated rather than defined, what we can do with them is limited to what is expressed by the axiomatisation.
A coinductive definition of semi-simplicial types
In this paper we propose a third enhancement of homotopy type theory, called Displayed Type Theory (dTT), in which it is possible to define and work with semi-simplicial types (and many other things). This type theory is inspired by the following idea for a coinductive definition of a type 
of semi-simplicial types:
Idea 2. A semi-simplicial type 
consists of:
-
• a type
, of
Zero simplices of , and -
• for each
-simplex of , a semi-simplicial type displayed over , called the
Slice of over .
, of
Zero simplices of 
, and
-simplex 
of 
, a semi-simplicial type 
displayed over 
, called the
Slice of 
over 
. To see why this definition is correct, let us first try to understand what a general semi-simplicial type 
displayed over 
consists of. The first part of this answer is that 
defines a type of 
-simplices of 
displayed over 
-simplices of 
, i.e. a family 
. Then, every displayed 
-simplex 
over a 
-simplex 
should define 
, a doubly displayed semi-simplicial type, for whatever this means.
Now, instead of working out the definition of a doubly dependent semi-simplicial type, let’s circle back and think geometrically. Semi-simplicial types should have families of 
-simplex types. If 
, then we write that:
is the type of 
-simplices in 
. Similarly, if 
is semi-simplicial type displayed over 
, then for 
, we write:
for the type of 
-simplices of 
displayed over the 
-simplex 
of 
. Putting this together, if we have two 
-simplices 
of 
, then we may form:
which is the type of 
-simplices in 
joining 
to 
.
It therefore stands to reason that 
should have a type of dependent 
-simplices living over the -simplices of 
. Thus if 
, then given dependent endpoints 
, we should get a type 
. The formula for this happens to take the following form:
which mirrors the formula for 
.
Then, putting all of this together again, if we have a 
-simplex 
, then we take 
. For 
, we have that 
. We thus get that:
In general, this pattern continues in higher dimensions and the process described lets us extract 
-simplex types.
We can visualise what’s going on in two different ways. The first visualisation shows how the 
-simplices of the slice of 
over 
live dependently over the 
-simplices of 
:
For example, if 
is a 
-simplex of the slice of 
over 
displayed over the zero simplex 
, then 
is a 
-simplex of 
joining 
. Similarly, suppose 
are 
-simplices of the slice of 
over 
displayed over 
, respectively, and 
is a 
-simplex of 
joining 
. Then if 
is a 
-simplex of the slice of 
over 
displayed over 
and joining 
, then 
is a 
-simplex of 
with the specified boundary.
Geometrically, we are using the fact that the 
-simplex is the cone of the 
-simplex. Thus, with the above definition, assuming we know inductively that every semi-simplicial type has a type of 
-simplices, then for every 
-simplex 
, the displayed semi-simplicial type 
has a type of ‘displayed 
-simplices’ over this. Such a displayed 
-simplex depends on 
(the cone vertex) as well as on an 
-simplex of 
(the base face, opposite the cone vertex) and thus can be viewed geometrically as an 
-simplex.
The second visualisation explains our formulas in terms of iterated slicing:
Note that, to form each successive slice, you have to provide 
simplex data points. The true dependent 
-simplices of a slice may then be viewed as matching objects. Hopefully this is sufficiently convincing for now; later we will give a precise justification.
As simple and appealing as this ‘definition’ is, it is not meaningful in ordinary dependent type theory. The intuitive claim is that it defines a type 
by coinduction, with 
and 
as destructors. For 
this is unproblematic (it is not even corecursive). However, the output of 
is not an element of the type 
being defined, as would be usual for a corecursive destructor of a coinductive type, but a ‘dependent element’, or ‘displayed element’, of 
over the input of 
. If we write 
for this putative family of ‘displayed elements’, the types of 
and 
are:
We would like to regard this as a sort of ‘higher coinductive type’. Just as a higher inductive type can have constructors involving not just elements of the type being defined but also paths therein, here we have a putative coinductive type whose destructors involve not just elements of the type being defined but also ‘displayed elements’ thereof. Thus, to make sense of this we need a type theory with a primitive operation 
associating to a type its family of ‘displayed elements’. As it turns out, the precise notion of 
that we require is a variant of unary internal parametricity.
External and internal parametricity
In general, by ‘parametricity’ we mean a statement that every type (perhaps subject to contextual restrictions; see below) is equipped with a relation (of some arity), and every function (subject to the same restrictions) preserves those relations. The original form of parametricity, such as in Wadler (Reference Wadler1989), is a meta-theoretic statement about type theory, in which the relations are meta-theoretic, and the contextual restriction is that it applies only to closed types and terms (those defined in the empty context). That is, in this ‘external’ parametricity, every closed type is given a relation on its closed terms, which is preserved by every closed function. For instance, in the unary case, given a closed function 
, if the 
-relation holds of a closed term 
, then the 
-relation holds of the closed term 
. Indeed, the fact that 
satisfies this condition is exactly the statement that the 
-relation holds of it, and is thus a special case of the ‘fundamental theorem of logical relations’ that every closed term satisfies the relation on its type. Semantically, external parametricity is obtained by interpreting type theory in a ‘gluing’ or ‘relational’ model.
By contrast, in type theories with fully internal parametricity such as Bernardy and Moulin (Reference Bernardy and Moulin2012), Bernardy (2015), Moulin (Reference Moulin2016), there is no contextual restriction, and the relations are internal (i.e. type families). In the unary case, this means for any type in any context, say 
, there is a type family that we will denote 
, and every function 
in any context lifts to a function 
between these type families. As in the external case, there is a formula for the relation of a function-type:
which says that so-called ‘computability witnesses’ for 
take computability witnesses of 
to computability witnesses of 
. Thus the statement that 
preserves computability witnesses is equivalent to saying it lifts to an element 
of this type, and is a special case of a general rule that every term 
lifts to a computability witness 
. (The punning of notation is intentional, and indeed consistent, as we will see: the type 
coincides with the term 
applied to elements of the universe.)
Such an internalisation of parametricity introduces the new possibility of iterating it, leading to 
, 
, and so on. Semantically, this means that each type must be interpreted by a cubical type (or set), where the arity of the relations determines the number of ‘boundary points’ on each side of the cubes, and the dependencies of each iterated relation on the previous ones supplies the faces of a cube. Our primary interest is in unary parametricity, where 
depends only on one copy of 
; in this case the semantics involves ‘unary cubes’ where each edge has only one vertex. (Geometrically, these can be thought of as powers of a half-open interval 
, or closed orthants 
in 
-dimensional Euclidean space.)
The rule lifting any 
to 
then implies that these cubical types must have degeneracies, taking any cube to a higher-dimensional one with some boundaries trivial. More surprisingly, it seems that to have good computational behaviour and semantic models, the cubical types must also include symmetries (transpositions): from 
, we can either directly obtain the type of 
as 
or first perform the computation 
and obtain its type as 
, and it these must be related by a symmetry operation.
The general advantages of internal parametricity over external are clear: we can reason about computability witnesses while staying within a single type theory, using a single proof assistant. Moreover, internal parametricity allows us to at least try to make sense of a type 
with the destructors (3). However, fully internal parametricity is not conservative over ordinary type theory: it is highly nonclassical, incompatible with axioms such as the law of excluded middle; and its semantics is not as general as we would like. We would like our type theory to be interpretable in any
$(\infty,1)$
-topos (generalising Shulman Reference Shulman2019), in such a way that our type 
is interpreted by a category-theoretic ‘classifier’ of semi-simplicial objects; but as we have just observed, the semantics of internal parametricity seems to live only in a category of cubical objects. Now the category of cubical objects in an
$(\infty,1)$
-topos is again an
$(\infty,1)$
-topos, so (using Shulman Reference Shulman2019) we can expect to interpret an internally parametric type theory in the latter; but the connection of this interpretation to the original
$(\infty,1)$
-topos is not clear.
Displayed type theory
Our solution is to use a less internally parametric type theory. We can think of internal parametricity as arising in three stages. The first stage is an external parametricity model, where types are interpreted by pairs 
. In this case 
maps one model to a different model, interpreting an ordinary type by a pair of types; this yields parametricity results as metatheorems. In the second stage, to make 
live in the same model as 
, we iterate this construction infinitely many times; now types are interpreted by semi-cubical types, with faces but not degeneracies. Finally, in the third stage we add a ‘degeneracy’ operation 
making every term parametric. This allows proving parametricity theorems inside the type theory, and semantically moves us from semi-cubical types to cubical ones.
The solution to our problem is to stop after the second stage. This is advantageous semantically because semi-cubical sets are presheaves on a direct category (i.e. covariant diagrams on an inverse category), and in this case the model construction is much more concrete. Specifically, in Shulman (Reference Shulman2015) it was shown that from any model of univalent dependent type theory inside of a type theoretic fibration category, one may form a derived model of Reedy fibrant presheaves on any direct category, with the type formers in the presheaf model constructed inductively in terms of those in the original model. In particular, in degree 
all the type-formers act exactly as they do in the original model. Thus, semantically we can be sure that all our constructions, including 
, specialise to something meaningful in an arbitrary
$(\infty,1)$
-topos.
dTT is a syntax corresponding to this model, which is likewise intermediate between external parametricity and fully internal parametricity: its parametricity primitive 
has a contextual restriction that is weaker than the ‘only closed terms’ requirement of external parametricity, but stronger than the ‘any context goes’ laxity of internal parametricity. We start by observing that in either cubical or semi-cubical sets, the semantically fundamental parametricity operation actually changes the context: given 
, one has 
, where 
augments 
by computability witnesses of all its variables. For cubical sets with degeneracies, we can deduce a version of 
that doesn’t change the context by substituting along a degeneracy map 
(e.g. this is the isomorphism between the ‘global’ and ‘local’ models of Reference Altenkirch, Chamoun, Kaposi and ShulmanAltenkirch et al. (2024)). But for semi-cubical sets this is impossible, so we have to bite the bullet and deal with context-modifying operations.
The notion of a non-binding operation that changes contexts is familiar from the realm of modal logic, where, to first approximation, a proof of necessity of some proposition, i.e. of 
, may only use necessary assumptions. Modalities in dependent type theory have previously been used to internalise meta-theoretic operations that don’t make sense in arbitrary contexts, such as the right adjoint to a 
-type in Licata et al. (Reference Licata, Orton, Pitts and Spitters2018), and we use them similarly here. Specifically, in dTT we have a modality 
that partially internalises the notion of ‘closed term’ appearing in external parametricity, and which restricts the domain of 
. Thus the only analogue of the above 
in dTT is 
. In particular, modal variables are protected from alteration by 
, so that we have 
, thereby avoiding the need for symmetry.
In fact, to emphasise further that dTT retains general semantics over an arbitrary
$(\infty,1)$
-topos, we will use a multimodal type theory Gratzer et al. (Reference Gratzer, Cavallo, Kavvos, Guatto and Birkedal2022, Reference Gratzer, Kavvos, Nuyts and Birkedal2021) with two modes, one for the original topos and the other for the topos of semi-cubical sets. These modes are related by modalities 
(the constant semi-cubical type), 
(the 
-cubes of a cubical type), and 
(the limit of a cubical type), and 
is a composite endo-modality. Only 
is necessary to formulate display, but the other modalities are also useful to have around: in particular, 
internalises the process of passing from the model in semi-cubical types to the original model in 
. For instance, 
is what corresponds semantically to the classifier of semi-simplicial objects in 
.
Furthermore, display itself may be thought of as a modality, albeit one that is indexed over the original type. Display falls into the new and yet underdeveloped framework of indexed modalities, such as the path types of cubical type theory (treated modally in Gratzer et al. Reference Gratzer, Cavallo, Kavvos, Guatto and Birkedal2022) and the identity types of the forthcoming Higher Observational Type Theory (HOTT) Altenkirch et al. (Reference Altenkirch, Chamoun, Kaposi and Shulman2024, Reference Altenkirch, Kaposi and Shulman2022). Moreover, analogously to those cases, we could formulate display either as an inert type-former defined by abstraction over an ‘interval’ (like path types in cubical type theory), or as an operation that computes on most other canonical type-formers (like identity types in HOTT). In this paper we make the latter choice, so that rules like eq. (4) are actually definitional equalities. Formulating such rules computationally is actually easier for dTT than for HOTT, due mainly to the lack of symmetry (although there is a tradeoff, since the presence of modalities is an extra complicating factor).
Until now we have been talking about semi-cubical types to make the connection with parametricity clear, but in the unary case there is an intriguing coincidence: the unary semi-cube category is isomorphic to the augmented semi-simplex category, with a dimension shift: the 
-cube corresponds to the 
-simplex. (To see this geometrically, note that the 
-dimensional orthant 
contains a standard face-preserving embedding of the 
-simplex, 
, including the augmentation case 
.) For this reason we refer to the two modes in our theory as the discrete mode 
and the simplicial mode 
. Thus, dTT can actually internalise semi-simplicial types in two ways: as the coinductive type 
mentioned above, and as the universe of types at the simplicial mode. The latter suggests that dTT could also be used similarly to simplicial type theory, with types at the simplicial mode treated as synthetic (augmented semi-) simplicial types. (Herbelin and Ramachandara Herbelin and Ramachandra (Reference Herbelin and Ramachandra2024) have also used iterated parametricity inside ordinary type theory to define augmented semi-simplicial sets and semi-cubical sets. The restriction to sets is because when not modifying the type theory, the higher coherences must be dealt with explicitly.)
(Note that an augmented semi-simplicial type can be viewed as a family of ordinary semi-simplicial sets indexed by the type of 
-simplices. Thus, our observation that augmented semi-simplicial types support a better internal language than ordinary ones is analogous to the observation of Riley et al. (Reference Riley, Finster and Licata2021) that parametrised spectra are likewise preferable to unparametrised ones.)
Displayed structures
Our terminology display for the operation 
is inspired by the fact that when applied to record types whose elements are algebraic structures, it produces displayed structures of the corresponding sort. Here a ‘displayed structure’ over a structure 
is a structure 
of the same kind with a structure map 
, but reformulated in terms of the corresponding family of fibres 
. Working with displayed structures rather than morphisms is a technique for enforcing definitional equalities on images in 
.
The most common displayed structure is a displayed category; here the terminology was introduced by Ahrens and Lumsdaine (Reference Ahrens and Lumsdaine2019). This arises from the record type of categories, defined in the usual dependently typed way (where we omit the axioms for concision):
We do not discuss record types (including 
-types) in this paper, but the extension of 
to them produces another record type whose fields have 
applied to them. For instance, from a 
-type:
We obtain:
Applying (4) and the similar rule 
, this becomes:
In a similar way, the above definition of the record type of categories yields:
Thus a displayed category over 
has a type of objects indexed by those of 
, types of morphisms indexed by pairs of objects-over-objects and by a morphism of 
, identity and composition operations on displayed objects and morphisms that lie strictly over those in 
, and similarly for the axioms.
As observed in Ahrens and Lumsdaine (Reference Ahrens and Lumsdaine2019), one use of displayed categories is to state definitions such as Grothendieck fibrations in terms of the existence of cartesian liftings strictly over any morphism in 
, without internalising definitional equality. Another is to construct categories and prove their properties in a modular way out of dependent pieces, just as we do for types using 
-types and more general records. It is ‘well-known’ by now that any sort of algebro-categorical structure has a ‘displayed version’ – for instance, displayed bicategories were used in Ahrens et al. (Reference Ahrens, Frumin, Maggesi, Veltri and van der Weide2021) to prove univalence modularly – but to our knowledge this has not previously been formalised. Our Displayed Type Theory (dTT) automatically generates the displayed version of any notion definable in type theory; hence the name.
Outline of the paper
The rest of this paper has three parts. In section 2 we describe the general syntax of dTT, including the modalities, the operation of display (in various different forms), and how they compute. We do not prove any canonicity or normalisation results, but we conjecture that they hold.
In section 3 we extend the syntax of dTT to define a type 
of semi-simplicial types. In fact, we obtain this as a special case of a general notion of ‘displayed coinductive type’, which is easier to work with abstractly, and also includes other important examples such as the type of semi-simplicial morphisms between two semi-simplicial types. (One might hope that it would also include the displayed versions 
, 
, etc., but this does not seem to be the case unless we add symmetry to our theory.) Then we explore a few applications, to make the point that this coinductive notion of semi-simplicial type is useful and practical.
Finally, in section 4 we consider the semantics of dTT. We will show that from any model 
of ordinary dependent type theory with countable inverse limits (roughly as considered in Kraus Reference Kraus, Herbelin, Letouzey and Sozeau2015), we can construct a model of dTT whose discrete mode is 
and whose simplicial mode is the category of Reedy fibrant augmented semi-simplicial diagrams in 
, and that this model supports displayed coinductive types including a type 
of semi-simplicial types. The underlying ordinary type theory of this model at the simplicial mode is an instance of the inverse diagram models of Shulman (Reference Shulman2015), Kapulkin and Lumsdaine (Reference Kapulkin and Lumsdaine2021), but we construct it more explicitly by hand so as to be able to verify the needed formulas for the additional operations of dTT.
Thus, although we don’t expect dTT to be conservative over ordinary dependent type theory, we can isolate exactly a kind of extra infinitary structure (namely, countable inverse limits) that yields a well-behaved theory for working with semi-simplicial types, which precisely includes the original model at one mode. In particular, by Shulman (Reference Shulman2019) any
$(\infty,1)$
-topos can be presented by a type-theoretic model topos, which is a model of type theory with countable inverse limits, and thus also yields a model of dTT. However, an object with the internal universal property of 
expressible in dTT has the potential to exist even in models that lack such infinitary limits, which may have implications for a notion of elementary
$(\infty,1)$
-topos.
2. Syntax
dTT is based on a multimodal type theory following Gratzer et al. (Reference Gratzer, Cavallo, Kavvos, Guatto and Birkedal2022, Reference Gratzer, Kavvos, Nuyts and Birkedal2021), with one mode for discrete types and another for (augmented semi-)simplicial types. It then adds a notion of ‘display’ at the simplicial mode that partially internalises unary parametricity. To state the computation rules for display, we need a generalised form of it that incorporates dependence on a telescope, so we also include a calculus of telescopes. (It would probably be possible to collapse this to dependence on a single type, using 
-types instead of telescope extension, as in Altenkirch et al. (Reference Altenkirch, Chamoun, Kaposi and Shulman2024), but this would be unaesthetic and less practical for implementation.)
2.1 The mode theory
We begin with a modal type theory based on the following 2-category 
:
-
• there are two modes (objects),
for discrete and for simplicial
-
• there are five nonidentity morphisms, forming hom-posets:
for discrete and 
for simplicial
-
composition is defined by the following tables (plus identity laws):
Intuitively, 
takes a discrete type and forms the constant (augmented semi-)simplicial type, while 
takes the 
-simplices of an (augmented semi-)simplicial type and 
takes the limit of an (augmented semi-)simplicial diagram.
The symbol 
is chosen to suggest a ‘diagonal’, and is sometimes used in category theory for any constant diagram functor 
, since this includes the ordinary diagonal 
as the case when 
. The symbols 
and 
are taken from classical modal logic, where 
denotes ‘necessarily’ and 
denotes ‘possibly’, although there is little formal connection except that 
, like necessity, is a comonad, and 
, like possibility, is a monad.
One verifies that the following adjunctions hold in 
:
Thus every morphism in 
, except for 
and 
, has a right adjoint.
2.2 The modal type theory
The basic structure of dTT follows MTT Gratzer et al. (Reference Gratzer, Kavvos, Nuyts and Birkedal2021), with judgments shown in Figure 1, where 
denotes a mode 
or 
. Following Coquand, we parametrise the type judgment by a universe level; the judgment ‘
’ generates these by 
and 
with a join 
. Flatness will be explained below.
Figure 1. Basic judgment forms of dTT.
The basic inference rules in Figure 2 formally define a Generalised Algebraic Theory (GAT) with these generating sorts. Accordingly, we present them using combinators for substitutions, including weakening; but in practice will generally use named variables and leave weakening implicit. Most rules are standard from MTT Gratzer et al. (Reference Gratzer, Kavvos, Nuyts and Birkedal2021) or MATT Shulman (Reference Shulman2023); for emphasis we have drawn boxes around the novel ones, which we now describe.
Figure 2. Basic rules of dTT.
In addition to functoriality of locks, we stipulate that 
and 
(but not 
) preserve empty contexts. (The rule for 
is redundant, since 
.) These equations mean that contexts do not have a unique presentation, but there are ways to select canonical presentations.
Semantically, each lock is left adjoint to its modality; thus 
and 
, so in particular 
and 
are also right adjoints. Intuitively, 
is an equivalence from the discrete types to the simplicial types that are empty in dimensions 
. This subcategory is a sieve, so 
is a parametric right adjoint, with a left adjoint defined on this subcategory, and which coincides with 
there. This subcategory is represented syntactically by the flatness judgment on 
-contexts, which are those containing a 
which is not to the left of any 
, as generated by the rules shown in Figure 2. The most important property of this subcategory is that 
acts as the identity on it; we represent this by the ‘exceptional’ key rule 
and its related variable rule.
We represent all three modalities in Fitch-style as in Gratzer et al. (Reference Gratzer, Cavallo, Kavvos, Guatto and Birkedal2022), Shulman (Reference Shulman2023), but with concrete parametric left adjoints rather than abstract ones. (We omit 
or 
, since their modalities can be obtained up to isomorphism by composing the others.) For 
and 
we use the actual left adjoints 
and 
with rules as in Shulman (Reference Shulman2023), while for 
the parametric right adjoint rules from Gratzer et al. (Reference Gratzer, Cavallo, Kavvos, Guatto and Birkedal2022) become a flatness assumption on the context.
2.3 Telescopes and meta-abstractions
Telescopesre suffixes of contexts not containing locks; they have a level that is no less than those of the types in them. Formally they are just another sort of the GAT, with judgments given in Figure 3 and rules given in Figure 4. We regard an operation on telescopes as ‘defined’ when we specify how it computes on empty telescopes and extensions by a type, such as is done in Figure 4 for the extension of a context by a telescope, 
. This operation is left-associative with the comma, so for instance 
means 
.
Figure 3. Telescope and meta-abstraction judgments.
Figure 4. Telescope rules.
A strict telescope (
) is one with no nontrivially modal variables. We treat the obvious map from strict telescopes to telescopes, and the operation of lifting a telescope to a higher level, as implicit coercions, using the same name for a telescope and its lifted and strict versions. We omit the obvious defining equations of these operations, and likewise for weakening by a telescope.
The judgement 
describes an ‘element’ of a telescope 
, which we call a ‘partial substitution’, thinking of it as a substitution 
that is the identity on 
. They are built out of terms, can extend ordinary substitutions, and uniquely determined by their components; the latter is ensured by detecting their equality by induced substitutions.
The judgement form 
defined in Figure 5 is more novel. We call it a meta-abstraction and think of it as saying that 
is a type depending on the variables 
in the telescope 
, i.e. belonging to a ‘framework-level 
-type’ 
. Accordingly, it is governed by binding and application, with a 
and 
-rule. We also regard 
as standing in for its own 
-type ‘
’, so such an 
can have its own terms belonging to it, with binding and application and a 
and 
-rule.
Figure 5. Meta-abstraction rules.
2.4 Décalage and displayed types
Semantically, the fundamental operation is shifting the dimensions of a simplicial type. In classical simplicial homotopy theory, this is called décalage:
The simplicial structure maps of 
are a subset of those of 
, while the unused ones assemble into a simplicial map 
. If 
is Reedy fibrant, then this map 
is a Reedy fibration, and so we can regard 
as the total space of a type dependent on 
, which we denote 
and call display. That is, 
is the fibre of the map 
; syntactically
Display is our version of the ‘logical relation’ assigned to every type by internal parametricity.
However, in contrast to fully internal parametricity theories, because we don’t have degeneracies in our cube category, décalage and display can only be applied in restricted contexts. In external parametricity, the logical relations apply only to types in the empty context; but our modalities allow us to say more generally that they apply to any ‘boxed’ type. Here by ‘box’ we mean not 
but the corresponding endofunctor of the simplicial mode, namely 
. Thus, informally display should have the type 
, with computability witnesses being assigned by a function 
. If we reformulate these without to 
-types, we obtain the following rules:
However, to compute with this, we need a version that incorporates dependence on a telescope to the right of the lock. The corresponding action on that telescope is décalage, which doubles the variables and groups each type with its displayed version, e.g.
The classical projection from décalage to the identity becomes an ‘evens’ substitution from 
to 
that throws away the elements of the displayed types (the primed variables above).
Now, the more general version of display can informally be thought of as:
However, this is not a well-behaved rule because the context of the conclusion is not fully general. We solve this by saying that general display acts on a meta-abstracted type, yielding the rules in Figure 6. Note that we put a superscript ‘
’ on variables in décalaged telescopes, and a prime on variables in displayed types and telescopes; these symbols are just a (mnemonic) part of the variable name. Our first, simple, sort of display is just an abbreviation for the case when the telescope 
is empty:
Figure 6. Décalage, evens, and display.
The more general display enables the rules ‘defining’ telescope décalage in Figure 7. When extending by a non-modal variable, we also extend by its displayed version, which depends on 
. Extending by a nontrivially modal variable is simpler: the modality has the form 
, hence semantically is a constant simplicial type. So we should have informally 
, and therefore 
is trivial, meaning we omit the displayed variables.
Figure 7. Computation of décalage and evens.
Finally, in Figure 8 we compute display on the basic type-formers, which as we discussed in section 1 acts like the identity types of HOTT. Note that the abstracting telescope changes as we compute, so these rules could not be stated for ordinary display alone. These rules represent the traditional behaviour of parametricity and logical relations on functions (a computability witness for a function says that it preserves computability witnesses) and universes (a computability witness for a type is a relation on it).
Figure 8. Computation of display.
2.5 Displayed telescopes
The rules given so far contain all the really important ideas of dTT, but in various contexts we find ourselves forced to introduce more general judgments and operations, involving telescopes dependent on other telescopes. In this section we briefly sketch these generalizations, which should be fairly intuitive; detailed rules governing them can be found in the extended version of the paper.
-
• The judgement
meta-abstracts a telescope over another telescope. -
• The operation
concatenates telescopes: it is a strictly associative -type. -
• The
-telescope of one telescope over another has the usual binding and application rules, and also computes on the structure of both or . -
• A meta-abstracted telescope can be dependently décalaged:
. This reduces to ordinary décalage on constant meta-abstractions, commutes with telescope concatenation, and computes on telescopes made out of types. -
• We also have display for telescopes: if
then . This always yields a strict telescope: its computation rules discard all nontrivially modal variables. -
• Compared to décalage, telescope display reorders the variables. For instance:
meta-abstracts a telescope over another telescope.
concatenates telescopes: it is a strictly associative 
-type.
-telescope 
of one telescope over another has the usual binding and application rules, and also computes on the structure of both 
or 
.
. This reduces to ordinary décalage on constant meta-abstractions, commutes with telescope concatenation, and computes on telescopes made out of types.
then 
. This always yields a strict telescope: its computation rules discard all nontrivially modal variables.
This difference is mediated by an ‘odds’ operation that picks out the elements of displayed types, and a ‘pairing’ operation that interleaves them together, such that evens and odds together form an isomorphism with pairing as inverse:
-
As with types, we have display for meta-abstracted telescopes, which décalages the domain telescope. This has a more general odds/evens isomorphism, and gives a way to compute display of a concatenated telescope.
-
Finally, either kind of display of a
-telescope computes by décalaging the domain and displaying the codomain. (It does not seem possible to consistently compute décalage of -telescopes.)
-telescope computes by décalaging the domain and displaying the codomain. (It does not seem possible to consistently compute décalage of 
-telescopes.)
Remark 5. We can now give a comparison to the ‘local theory’ from Altenkirch et al. (Reference Altenkirch, Chamoun, Kaposi and Shulman2024), with our telescopes and telescope concatenation replacing their types and 
-types. Our décalaged telescope 
from section 2.4 corresponds to their 
, while the meta-abstracted analogue corresponds to their 
. Our décalaged partial substitution 
from section 2.4 corresponds to their 
(although with an added modal lock), while its meta-abstracted analogue corresponds to their 
(with modal lock). We don’t have their 
represented explicitly, but one of their rules says it is equivalent to 
in a constant family. Our 
is their 
(in the unary case, so 
), and our modal guards make their 
unnecessary.
3. Semi-Simplicial and Displayed Coinductive Types
Recall from the introduction that our primary goal in formulating dTT (at the moment) is to have a type theory in which we can make precise our coinductive definition of the type 
of semi-simplicial types. In this section we give that definition, along with a more general notion of ‘displayed coinductive type’, and explore some applications.
3.1 Semi-simplicial types
In a proof assistant implementing displayed type theory, one would expect to define semi-simplicial types as an instance of a general codata declaration. In addition to the display primitive, this requires an enhancement of the usual syntaxes for coinductive definitions that allows the coinductive input of each destructor to be named and referred to in its type. For instance, using an Agda-like syntax for records we might write the following:
In fact, this is already possible: the second author has implemented an experimental proof assistant called Narya for ‘observational’ higher type theories like dTT and HOTT. Narya doesn’t yet include the modalities of dTT, but it does have a general notion of coinductive definition that includes 
, with the following syntax:
Note that Narya regards 
and 
as fields, written with postfix syntax, rather than functions written applicatively. Also it requires parentheses around 
, for reasons we’ll mention later.
A sufficiently broad framework encompassing all such ‘displayed coinductive’ definitions (let alone a full specification of Narya) is beyond the scope of this paper, but we will describe one general paradigmatic class of them, analogous to W-types as paradigmatic inductive types and M-types as paradigmatic coinductive types.
We begin by discussing the motivating example of 
in more detail, starting with its type formation law and destructors. Of course, since 
is a sort of ‘universe’, its elements consisting of types, it must also be parametrised by a level.
Note that the destructors are defined as terms belonging to ‘meta-abstractions’ as introduced in section 2.3. We have chosen this over the more common method of supplying the arguments in premises, for example:
because it makes it easier to compute 
of them:
This suggests that the family 
should behave as though defined by computing 
on all of the destructors of 
, regarded as functions:
Unfortunately, as we will see this is not actually possible in the version of dTT we have described in this paper, but it is the correct intuition.
Now suppose we want to construct a function mapping into 
from some telescope. We first think purely in terms of code, written in the style of Agda-esque copattern matching, with the goal of writing down something that can conceivably be justified:
Here, suppose that 
. If we think of 
as a state space and 
as a state, then the above definition suggests that we are able to define 
provided that we are able to provide two ingredients. First, we need a way of extracting 
, a type of 
-simplices, from a state 
. Second, we need a way of extracting 
, a dependent section of 
over 
, from a state 
and a 
-simplex 
. This suggests that a reasonable coinduction principle for 
is the following:
and that its computation rules should be:
Now 
belongs to 
, and one might hope to compute this using a corecursor for 
. More generally, let us try to work out the coinduction principle for defining 
, where 
. Using the previous methodology, we start with reasonable-looking code:
However, this doesn’t actually typecheck, because we have
Thus, there is an index ordering mismatch that seems to prevent us from writing down a coinduction principle for 
. The solution is to extend dTT with symmetries; this would allow us to line up the 
indices and impose the definitional equality 
as a corecursor premise. In fact, Narya implements symmetries, so this is possible there:
Narya doesn’t require the 
on 
and 
, since that is implicit in the type 
of 
. Symmetries are also the reason for Narya’s parentheses: its 
is an instance of a general operation that also includes all higher-dimensional symmetries represented by numerical permutation notations, such as the 
used here for a transposition.
In this paper we have chosen to develop a theory without symmetries, so we must abandon this approach. Instead we can leave 
as a stuck form, but specify how to compute 
and 
on it. The main idea is that if we define:
then we can compute display on each line of this definition to obtain:
Thus we obtain the computation laws:
If we think of pairing as a 
-telescope, 
, then specialising this to 
gives 
. On the other hand, taking 
-telescope décalage gives 
. As we shall note in Section 3.3, display and décalage do not commute, so we must be careful to distinguish these operations. In the case of a single type, 
reduces to concatenation, and more generally, 
reduces to interwoven concatenation for each variable in 
.
We can iterate this to obtain:
The situation is not unlike (co)pattern matches in Agda (and Narya), which define new normal forms rather than expanding to a first-class introduction or elimination form; such names only reduce when their defining patterns occur.
3.2 Examples of semi-simplicial types
Of course, simply defining a type of semi-simplicial types is only the first step: we also want to be able to work with such things conveniently. Developing a full theory of semi-simplicial types is beyond the scope of this paper, but in this section we give a few examples to suggest that this may be possible with our definition of 
and its corecursion principle. We will use Agda-esque copattern-matching, and assume that our type theory has plenty of other structure rather than the bare-bones version of dTT that we have studied formally in this paper. Many of our examples have been formalised in Narya. (However, in attempting to go beyond the examples here, we have found that symmetries are generally unavoidable.)
3.2.1 The singular semi-simplicial types
Display is independent from any notion of propositional equality, whether that be Martin-Löf, cubical, or observational. However, we now want to define a semi-simplicial type that arises from the
$\infty$
-groupoid structure of a type in HoTT, and for concreteness we will use a cubical notion of equality. In Cubical Agda notation, we have:
Thus, it is well-typed to declare the latter to be the display of the former:
With this given, the singular semi-simplicial types are defined by corecursion. Rather than write this explicitly using the corecursor from section 3.1, we use a copattern-matching syntax, including a ‘displayed corecursive call’ 
:
A calculation then yields:
3.2.2 Nerves of categories
Let 
denote the type of 1-categories, defined as a record inside dTT (extended by record types), and recall that in section 1 we computed 
to consist of ‘displayed categories’ in the usual sense Ahrens and Lumsdaine (Reference Ahrens and Lumsdaine2019). Thus we can define:
Here by 
we mean the coslice category 
, regarded as a displayed category over 
via the forgetful functor. Note that defining 
globally in dTT automatically induces the definition of 
. A similar idea works for bicategories, and any other kind of category that has a displayed (co)slice.
3.2.3 Topological singular complexes
For any type 
of ‘topological space’ definable inside of dTT as a record, we have a displayed version 
. In some cases an element of the resulting 
is more general than an 
with a map 
; but at least from such a 
we can construct its fibers as a displayed space. Thus, as long as we can construct, for any 
, a space of ‘continuous paths in 
starting at 
’ with an endpoint projection down to 
, we can make it a displayed space 
over 
, and use this to construct the singular semi-simplicial types:
3.2.4 Fibers and higher spans
As we will see in section 4, semantically each type at mode 
is already an augmented semi-simplicial type. Thus, we expect that if we fix a 
-simplex in such an augmented semi-simplicial type, we should obtain an (unaugmented) semi-simplicial type as its ‘fibre’. And indeed, we can define this operation:
Note that 
is required to be modal so that we can take display of it. Then we have, for instance, if 
and 
, that:
and as a last example:
In particular, if we let 
be a universe and 
be a unit type, we have:
Thus 
is the semi-simplicial type of types, spans, and a sort of simplicial ‘higher spans’ that could also be called ‘heterogeneous simplices’. More generally, 
for any type 
consists of types, spans, and simplicial higher spans indexed by 
.
3.2.5 Operations on semi-simplicial types
We can also use corecursion to define operations on semi-simplicial types that are essentially levelwise. For instance, any two semi-simplicial types have a product:
Here we treat the non-displayed arguments of 
as implicit – its full type is:
There is an empty semi-simplicial type. Note that the 
case can be omitted, since one of its arguments would belong to the empty type 
.
Similarly, there is a trivial one:
We can also take the product of any family of semi-simplicial types indexed by a discrete type. The discreteness of 
means that it doesn’t need a displayed version in 
.
However, some things do not seem possible with our current theory. For example, the disjoint union of semi-simplicial types should certainly have the disjoint union of 
-simplices, but the slice over a 0-simplex should come only from one side: 
should morally be 
. But 
belongs to 
, whereas 
must belong to 
. We need to take its disjoint union with an ‘empty semi-simplicial type displayed over 
’; but what is that? We defined a ‘global’ empty semi-simplicial type above, and it seems intuitively that we should be able to define a ‘constant’ version of this displayed over 
. But as noted in section 3.1, without symmetry it does not seem possible to formulate a useful corecursor for 
, so it is unclear how to define ‘constantly displayed’ semi-simplicial types. There are many other examples of this sort that suggest that further work in this direction will require the inclusion of symmetries.
3.3 Displayed coinductive types
Generalising the discussion of 
, we now formulate a fairly general notion of ‘indexed displayed coinductive type’. It depends on a telescope 
of ‘non-uniform parameters’, and every element of it has a ‘head’ belonging to some specified type family 
and a ‘tail’, depending on a telescope of parameters 
, and belonging to the displayed version of the coinductive type itself. The parameters of this displayed version of the very type being defined are 
, which we can assemble provided that the data of the old parameters 
, the head 
, and the new dependencies 
, are sufficient to extract a section 
. The idea is analogous to an ‘indexed M-type’, but with the output of the tail being displayed, and with 
being a telescope rather than a simple type. The pseudo-Agda corresponding to this would be:
We can thus write down the formation and introduction rules for 
as follows:
Note that the universe level of 
is governed by those of 
and 
, but does not depend on the level of the telescope of non-uniform parameters 
.
Following the example of 
, we will begin by attempting to write down a reasonable template for a coinduction principle. In the same module context, we can attempt to map into a 
type from a length two context as follows:
The types that we have are then:
Thus there is a non-trivial condition that needs to be imposed for this definition template to be well typed. Fortunately, unlike in the case of 
, we generally have terms lining up in the sense that the terminal 
terms align. We get the following rule:
This comes with the following computation rules:
Now we can define 
as a particular indexed displayed coinductive type (which happens to have trivial indexing). In fact, it is the ‘universal’ such instance, where the family 
indexed by 
is the universal family 
indexed by 
. This may be compared with the fact that the 
-type of 
is the type of ‘presentations of well-founded sets’ Aczel (Reference Aczel, Macintyre, Pacholski and Paris1978), while its 
-type is the type of ‘presentations of ill-founded sets’ Lindström (Reference Lindström1989).
Then we can deduce the rules for 
, defining 
and 
and 
.
The problem of giving a corecursion rule for 
carries over to the general case. Specifically, just as 
of a 
-type is another 
-type and so on for records and ordinary coinductive types, we’d like to compute 
of a 
to be another 
, with something like this:
To see whether this is well-typed, observe that we have:
whereas the 
of the resulting 
must lie in 
. Thus, in particular, we need to compare 
, where 
. In the case of a one-type telescope 
, this becomes:
Unfortunately the last two are not the same! This is not just about ordering the variables in a telescope; although the second and third arguments of 
both lie in 
, it need not be symmetrical with respect to those arguments. So again we see that without adding symmetry to the theory, it seems we can’t give a general corecursor for 
, and hence we can’t compute 
to something more primitive.
In Narya, which does have symmetries, the analogues of 
are again displayed coinductive types, which therefore permit copattern-matching with displayed corecursive calls; although specific constructions like 
do not actually compute to anything until their copatterns apply, as with ordinary corecursive definitions. However, since Narya presently lacks modalities, we cannot formulate in it a generic displayed coinductive such as 
with arbitrary parameters such as 
, only specific examples.
3.4 Examples of displayed coinductive types
We continue our exploration of the theory of semi-simplicial types from section 3.2, now using the general notion of displayed coinductive type. As in section 3.2, we will use Agda-esque codata and copattern-matching definitions, and assume that our type theory has plenty of other structure.
We have already noted that 
is in some sense the ‘universal’ (unparametrised) displayed coinductive type, whose determining family 
is the universal one 
. Moreover, it seems likely that in order for an unparametrised displayed coinductive type to be interesting, the types 
and 
must have nontrivial display structure, i.e. they must not be discrete. But the simplicial universe 
is the primary source of types with nontrivial display, just as the universe in homotopy type theory is a primary source of types with higher homotopy structure. (In section 5.9 we will speculate about a notion of ‘display inductive type’ analogous to higher inductive types, which are the other source of higher homotopy structure in homotopy type theory.) For these reasons, we do not have a lot of interesting examples of other unparametrised displayed coinductive types, but there is at least one: augmented semi-simplicial types.
3.4.1 Augmented semi-simplicial types
If we simply omit the 
input of 
in the definition of 
, we obtain a definition of augmented semi-simplicial types:
We can extract the types of low-dimensional simplices from an 
:
and so on. Now we can observe that the construction 
of section 3.2.4 factors through 
via a pair of maps, both defined by copattern-matching:
3.4.2 Pointed semi-simplicial types
More interesting examples of displayed coinductive types have nontrivial parametrisations, often involving more semi-simplicial types. For instance, we can define the structure of a pointing on a semi-simplicial type displayed-coinductively:
We then have, for 
, that:
and so on. That is, an element of 
equips 
with a ‘fat point’, i.e. a chosen 
-simplex 
that comes with all of the higher ‘degenerate simplices’ that one would expect to be associated to 
if it were in a simplicial set rather than a semi-simplicial one.
3.4.3 Morphisms of semi-simplicial types
With a double parametrisation, we can define a type of morphisms of semi-simplicial types:
As usual, we can unravel this a few steps to see what it looks like. 
is a function between types of 
-simplices, which we may denote 
. At the next dimension we have:
which is to say that:
We may denote this function by 
, and go on to extract a function 
between types of 
-simplices and so on. We expect other basic operations on semi-simplicial types to be internalisable in a similar way.
4. Semantics
We now construct, from any model 
of ordinary dependent type theory with limits of countably infinite towers of types, a model of dTT in which 
sits as the discrete mode.
Remark 6. For simplicity and space, we will omit the type-former 
. It should be possible to model 
as well, as long as 
has a unit type, but we leave the details for the future. We will still model 
-modal variables and function types such as 
.
In section 4.1 we extend the semantics of dependent type theory to telescopes and meta-abstractions, plus countable infinite limits. Then in section 4.2 we use these limits to construct a model in augmented semi-simplicial Reedy diagrams. This is essentially an instance of Shulman (Reference Shulman2015), Kapulkin and Lumsdaine (Reference Kapulkin and Lumsdaine2021), but our explicit version builds in display and décalage. In section 4.3 we add modalities, and in section 4.4 we discuss the general notion of model of dTT. Finally, in section 4.5 we construct displayed coinductive types in these models, including 
.
4.1 The semantics of dependent type theory
4.1.1 CwFs and natural models
As our notion of model for type theory we will use categories with families (CwF) Castellan et al. (Reference Castellan, Clairambault, Dybjer, Casadio and Scott2021), in their equivalent presentation as natural models Awodey (Reference Awodey2018), with universe levels:
Definition 7. A natural model with levels is a category 
with a terminal object and a family of algebraically representable natural transformations 
of presheaves on 
.
We view a presheaf 
on 
as a hypothetical judgment form, and elements of 
as 
-judgments in context 
. For instance, 
is the judgment 
, which we will also write as 
to emphasize the dependence of 
on 
. We denote the functorial action of 
on types 
and terms 
by 
and 
:
or equivalently by taking the formal variables 
more seriously and ‘applying’ 
to them:
This can be formalized using the internal logic of presheaves. Similarly, plain sets such as the objects and morphisms of 
can be viewed as ‘absolute’ or ‘context-less’ judgments.
The representability of 
gives, for each type 
, a context extension 
, with a family of bijections, natural in 
:
By the Yoneda lemma, the forwards direction of this bijection is determined by a fundamental context projection or parent map, and a zero variable term:
Then the forwards direction of the bijection sends a substitution 
to the pair 
, and its inverse is a substitution extension operation:
such that for 
and 
and 
, we have:
As a corollary of this, the following diagram is a pullback:
We call the map in the top row above the weakening two of 
by 
, and write it 
. Note that in Altenkirch et al.(Reference Altenkirch, Hofmann and Streicher1995), they define 
for 
.
Finally, instead of 
, we may write 
, and so on. Thus, in particular, 
is just 
. As before, this can be justified formally as an interpretation in the internal type theory of 
.
4.1.2 Polynomial endofunctors
Each representable map 
induces a polynomial endofunctor 
of 
, where for any presheaf (i.e. judgment) 
, elements of 
(i.e. 
-judgments in context 
) are bijective to pairs of a type 
in context 
and an element of 
.
Similarly, 
contains 
-judgments in a context 
. Thus, it is the polynomial functor associated to the map 
where the fiber of 
over 
consists of a pair of terms 
.
For example, 
represents families of types of level 
indexed by a type of level 
. Thus the formation rule of 
-types:
is represented by a morphism 
. Other types and term rules can also be represented this way; see Awodey (Reference Awodey2018). We will assume 
-types and universes.
4.1.3 Telescopes
Definition 9. A natural model with levels 
has telescopes if it is equipped with:
-
• Another family of representable transformations
. -
• Morphisms of polynomial functors
, yielding empty telescopes. -
• Morphisms of polynomial functors
whenever , yielding the extension of a telescope by a type. Note is the polynomial functor of a map with codomain , the presheaf of types in a context extended by a telescope. -
• The rules
and from section 2.3 hold, along with their analogues for partial substitutions. These are equalities of objects of , and only make sense if and are algebraically representable. -
• A morphism of polynomial functors
, yielding the concatenation of telescopes mentioned in section 2.5, such that the relevant rules hold.
.
, yielding empty telescopes.
whenever 
, yielding the extension of a telescope by a type. Note 
is the polynomial functor of a map with codomain 
, the presheaf of types in a context extended by a telescope.
and 
from section 
, and only make sense if 
and 
are algebraically representable.
, yielding the concatenation of telescopes mentioned in section Additionally, it has
$\Pi$
-telescopes if there is a pullback square:
satisfying the computation rules alluded to in section 2.5.
This represents the rules from section 2.3 in the non-modal case. By using presheaves, it implicitly includes substitution into telescopes that commute with the other operations.
As for contexts, the rules do not require that every telescope is obtained by successive extension of the empty telescope by types. Indeed, there is no way to assert such a thing in a Generalised Algebraic Theory. But it holds ‘admissibly’ in the initial syntactic model, and any CwF can be extended with telescopes in this way.
Theorem 10. Any natural model with levels can be equipped with telescopes. If it has 
-types, it also has 
-telescopes.
Proof. We define 
to be the map such that 
Thus an element of 
is a tower of 
types over 
of level 
, and similarly for terms. We define context extension in the obvious way by iterating context extension by types, and the operations and equations are immediate. Similarly, we define 
-telescopes by using the rules for computing them on extended telescopes.
4.1.4 Meta-abstractions
Because meta-abstractions are not ‘reified’ as types, they do not require any additional structure. Specifically, the rules for the judgment 
simply say that it is the object 
that classifies types indexed by a telescope, and similarly elements are classified by 
. Thus meta-abstractions of types are classified by the map 
, and similarly meta-abstractions of telescopes are classified by 
; thus any natural model with telescopes also has meta-abstractions of types and telescopes. Semantically, we don’t need to discuss meta-abstractions explicitly, since 
is equivalent to 
, and so on. Thus we will generally talk only about types and telescopes in contexts.
4.1.5 Infinite telescopes
We now define a new judgement 
whose elements are ‘infinite telescopes’. As with meta-abstractions, this is not (yet) introducing new structure on a CwF, rather it is a definition that can be made in the presheaf category of any CwF. The idea is that an infinite telescope consists of an infinite sequence of types each dependent on all the previous ones:
where 
for all 
. Formally, we define this along with its finite approximations:
so that we can say that in general 
is a type in context 
. In syntax, this means we give the following bidirectional rule with infinitely many premises:
Similarly, the judgement 
of infinite partial substitutions is characterised by a bidirectional rule:
Substitution extends level-wise to infinite telescopes and their partial substitutions. Categorically, these rules mean the map 
is the limit of the sequence:
where 
is such that 
Thus 
is the identity functor, 
is the identity map of the terminal object, and 
discards the last type in a telescope of length 
(there being no other way to get a telescope of length 
).
4.1.6
-limits
-limits Finally, we define the structure of infinite (sequential, Reedy) limits on a CwF. These are an ‘infinitary rule’ (i.e. a non-elementary structure) that is not part of dTT or any implementable type theory, but we will use them to build our intended models of dTT. Syntactically, they are essentially just a kind of 
-type of an infinite telescope.
Definition 11. A CwF with levels has 
-limits if it is equipped with pullback squares:
In syntax, this means we have the following structure and properties. First, having a merely commutative square as above gives the following rules:
Second, we have restriction operations:
We require that 
is derived from 
via:
and that the following computation and uniqueness rules hold:
Of course, all these constructions must also be stable under substitution.
4.2 The simplicial model
Fix a natural model with all of the structure described above, which we call the discrete model (
). We will construct a new model called the simplicial model (
), by first constructing the truncated simplicial models (
) for 
.
Let 
be the type of binary digits, 
. For 
, let 
be the type of length 
binary sequences of which exactly 
are 
. The augmented semi-simplex category 
has objects 
and morphisms 
. The identity 
is all 
s, and the composite of 
and 
replaces the 
s in 
with the digits of 
; e.g. 
. Note that each representable 
is finite.
We write 
and 
for left appending digits. Then:
Note that by the second rule, along with the fact that 
, the assignments 
and 
define an endofunctor of 
.
4.2.1 The truncated simplicial models
For every 
, let 
be the full subcategory of 
on the 
with 
. Thus 
is empty, while 
is terminal. The 
-truncated simplicial model 
has underlying category 
. For an object of this category, written 
, we have 
whenever 
, where 
is terminal; and if 
then 
, contravariantly functorially. We sometimes write 
for 
. A morphism 
consists of 
for each 
such that 
for any 
.
The 
are related by truncation and décalage functors and a natural transformation:
We will now construct the type-theoretical/fibrant structure of the models 
with a large set of mutually inductive definitions. Our first goal is to define the judgement:
Intuitively, such a simplicial type 
consists of discrete 
-simplex types for 
:
Note that 
denotes the 
-component of the presheaf 
(a 
-object), while 
is an atomic variable name in this object. And the type of 
is well-typed because the outer rectangle of the following diagram is a distinguished pullback:
We will write the type declarations of 
generically as:
Here 
is a telescope consisting of the ‘boundary’ of an 
-simplex, also known as the Reedy ‘matching object’ of an augmented semi-simplicial type. For example:
Similarly, the term judgement
will be defined to consist of the data:
Similarly to before, we will write this generically as:
where 
acts by the lower-dimensional parts of 
on the boundary of 
.
Therefore, we need to mutually define these matching telescopes and matching substitutions. To define these, in turn, we need a truncated form of display, which takes an 
-truncated semi-simplicial diagram 
to an 
-truncated one dependent on 
. Since we have no modal locks available yet, this version of display must be in a totally décalaged context. Finally, to make the context extension a presheaf, we need the actions of morphisms in 
on matching telescopes and on types. The complete list of judgments and operations being defined in our first mutual induction is shown in Figure 9.
Figure 9. Structures defined mutually for truncated simplicial types and terms.
The inductive definitions of these judgments and operations are shown in Figure 10. The judgments are defined by bidirectional rules, meaning they are presheaves of ordered pairs, with the premises giving notation for the two components. We also have to define the substitution action of these presheaves in terms of these components. Similarly, after defining context extension, we have to specify the functorial action to make the extended context an augmented semi-simplicial object. The matching telescopes and matching substitutions are defined inductively on 
; display is defined in terms of components; and the simplicial action on matching objects is defined based on the simplicial morphisms.
Figure 10. Mutual inductive definitions for truncated simplicial types and terms.
There are also many equations to verify as part of the induction: everything has to be well-typed and stable under substitution, and we need computation rules for display:
and functoriality of the 
operations:
Some of these proofs can be found in the extended version of this paper.
Now to make 
a natural model, we need fundamental projections and variables:
We define these inductively:
To make this well-typed, we have to simultaneously prove eq. (8) as well as:
These proofs can also be found in the extended version.
4.2.2
-types
-types Assuming the discrete model has 
-types, we construct them in 
inductively. Assuming 
-types at prior levels, we have the following two types in the same context:
In addition to the definition of 
-types, we include in the induction the statement that these two types are equal. Using this assumption, we can give the base definitions at 
as
and continue inductively to make the whole construction level-wise:
The mutual proofs that with these definitions the above two types are equal, and that the 
and 
laws hold, can be found in the extended version.
4.2.3 Universes
Denoting universes of the discrete model by 
, we construct universes in 
inductively, assuming them at all prior levels. We include in the induction that:
Assuming this inductively, for 
we define:
and at the inductive step we make things level-wise:
The proofs of eqs. (16)–(18), and that 
and 
are mutual inverses, can be found in the extended version.
4.2.4
-limits
-limits Assuming 
-limits exist in 
, and inductively in 
, we first construct the display of infinite telescopes and partial substitutions in 
. These operations have types:
and we define them by:
We include in the induction the following theorems:
Assuming this inductively, the definitions are:
The proofs of all the necessary equations can be found in the extended version.
4.2.5 The simplicial model
Finally, we obtain the untruncated simplicial model 
by taking a limit. We first define a tail-cutting truncation functor, extend décalage to an endofunctor, and 
without truncation:
Now we define the types and terms in 
to be compatible towers of types and terms in the 
. In syntax this can be expressed by the following infinitary bidirectional rules:
We also define:
All the above constructions in 
extend to 
levelwise, since they are preserved strictly by the 
. In particular, display is modified in the absence of truncation:
The computation rules for display similarly hold in 
when modified to exclude 
.
4.3 Modalities
4.3.1 Pieces of the triangle modality
The modality 
is supposed to construct a constant (augmented semi-)simplicial diagram, while its left adjoint 
picks out the object of 
-simplices. Both are determined levelwise by their behaviour on truncated diagrams. (Recall that we will not be modelling the modality 
on types itself.) We begin by defining a functor 
via:
Then we construct modal extension for 
in 
:
The construction is inductive, as before, and the induction includes the following equations:
Assuming this, the definitions are:
We next have fundamental context projections and zero variables:
The definition is again inductive, and the induction includes that:
Assuming this, the definitions are:
Finally, we construct modal 
-types:
This construction follows the pattern of the non-modal truncated case and is performed level-wise. We will inductively assert the following formulas for display:
In dimension 
we set:
Then we inductively define:
As usual, the verification of many identities has been omitted.
Finally, we check that the truncation functors 
preserve all the operations defined above. Therefore, we can define the untruncated operations 
and 
on 
, with modal context extension 
and modal 
-types, simply by acting levelwise on each 
.
4.3.2 Pieces of the box modality
The box modality is not determined levelwise by operations on truncated diagrams, but we can still construct it using truncated data. We start with a truncated lock functor 
that constructs a constant simplicial diagram:
We the define four new operations, primarily 
which is like a truncated version of 
, taking the limit of a truncated diagram, but yielding a finite telescope rather than a type:
These will satisfy the inductively proven property that for 
:
For 
, the term 
is trivial, since it lives in the terminal CwF. We also set:
Note that, in general, since 
is a simplicial term, we may form its matching substitution:
For 
we then inductively set:
The second line is well typed by the inductive hypothesis and makes the next case of the hypothesis clear. Also, note the 
substitution in the fourth line. The top-dimensional simplicial value of a boxed variable accesses the last component of the modal context extension, whereas lower dimensional simplicial values search further back in the linear context.
The untruncated 
similarly constructs a constant presheaf. Note that 
, and 
is an identity; thus we will omit writing these. We now define:
We then use the constructions above to construct a modal type former:
(Recall from section 4.3.1 that 
.) In order to form these, we will take an 
-limit of sequences 
or 
obtained from the 
-simplex levels of 
or 
:
These are defined as follows:
We then define:
We define the eliminator by:
and then check the computation laws.
4.3.3 The extended simplicial model
Syntactically, the context 
is supposed to be flat. A naïve approach to modelling this would add initial objects to a discrete context. Instead we define an extended simplicial model 
, built from a copy of 
(the flat contexts) and 
(the non-flat contexts). Its objects, morphisms, types, and terms are generated inductively by the rules in Figure 11. Up to isomorphism, its underlying category 
is the ‘collage’ or ‘cograph’ of the functor 
; the rules of Figure 11. implement this by labelling everything based on its origin. We define the action of substitutions as follows, discarding higher data in the flat case:
Figure 11. The extended simplicial model.
Extension of contexts operates by passing under the inclusion:
Note that in the last case, if we have 
then 
. To form the extension 
, we must give 
. Such an 
has type 
and must be of the form 
.
To define (non-modal) 
-types and universes in 
, we reduce to the respective constructs in 
and 
, depending on whether or not the context is flat:
The definitions of 
, and 
are similar. For stability under substitution, we have to additionally consider flat substitutions; this works because everything agrees with its discrete version in dimension 
. In the rest of this section, we show how 
and 
can be made into a model of all of dTT (except for the type-former 
).
4.3.4 Locks and keys
The definition of 
is tailored to 
. Putting this together with the 
and 
defined on 
in Sections 4.3.1 and 4.3.2, we now define a 
-functor 
, where 
denotes the 
-category obtained by reversing both 
and 
cells. On modes, we have:
To define this 
-functor on modalities, we extend the prior definitions of locks to 
:
We then define the evident composites:
Finally, it is easy to check that 
and 
define identity functors. It follows that we have a contravariantly functorial assignment:
To define this 
-functor on 
-cells, we define the key transformations. We have 
, 
, and 
, which corresponds to the following natural transformations:
We also have 
. These are contravariantly functorial:
One checks whiskering identities to verify that 
defines a 
-functor 
.
4.3.5 Modal types
We are modelling two modal type formers, with their introduction and elimination rules:
We have simplified the statement of 
, using the fact that 
means 
is of the form 
, in which case we have 
by definition. The key 
does not arise from a natural transformation and is only defined when 
, in which case 
. Thus, this can be seen as a proof-relevant pattern match along 
.
The definition of the 
rules shuffle around discrete information and inclusions:
For 
, we fall back to our prior construction for the type former and introduction rule:
while for the eliminator, we split on whether or not 
is flat:
where the discrete case above is as follows:
It is defined by:
4.3.6 Modal variables
We have the following rules for extending a context and substitution modally:
The case of 
is defined as follows, splitting on whether or not 
is flat:
Following this, the rest of the definitions say that the case of modal extension reduces to extension by a variable or term of modal type:
Each of the context extension operations comes with a notion of parent maps and variables:
These are defined as follows:
4.3.7 Modal -types
-types Finally, modal 
-types must satisfy the following rules:
For 
we define:
The other cases reduce to functions of a modal variable:
The cases of 
and 
are similar.
4.4 Semantics of dTT
Since our syntax was presented as a Generalised Algebraic Theory, there is an immediate notion of model, namely an algebra for that theory (with algebraic syntax being the initial model). We now reformulate this in more familiar category-theoretic terms.
4.4.1 Modal structure
The general multimodal type theory MTT was presented algebraically in Gratzer et al. (Reference Gratzer, Kavvos, Nuyts and Birkedal2021). For a general mode 2-category 
, the starting point is a modal context structure, which is a 2-functor 
, where 
denotes reversal of both 1-cells and 2-cells. The image of a mode 
is the category 
of contexts and substitutions at that mode, and the image of a morphism 
is the lock functor 
, which we write postfix, 
. It is a modal natural model if each 
is equipped with a morphism 
in its presheaf category such that for any 
the morphism 
is representable. (In particular, taking 
, we see that each 
is an ordinary natural model, hence a CwF.) This notion encapsulates all the rules for building contexts and substitutions from section 2.2 except for those that refer to flatness of contexts.
Now, specialising to our mode 2-category from section 2.1, the rules of section 2.2 say that the flat contexts form a full subcategory of 
that contains the image of 
. In our algebraic theory we take as primitive the derived rules that a context is flat if and only if it admits a substitution to 
, and in that case the latter substitution is unique. Thus, semantically, 
is subterminal and the flat contexts are the slice category 
. Since the unit of the adjunction 
is an identity, 
is fully faithful, and on its image it has 
as an inverse and therefore also a left adjoint. In addition, since flat contexts are fixed points (up to isomorphism) of 
, when 
is flat there is a bijection 
, and so the modal comprehension 
is isomorphic to an ordinary one 
. This justifies the special variable rule and key substitution. Thus, the following definition encapsulates the judgmental structure of our modal type theory.
Definition 19. A dTT context structure is a modal context structure 
in the sense of Gratzer et al. (Reference Gratzer, Kavvos, Nuyts and Birkedal2021), where 
is as in section 2.1, such that 
is subterminal and the slice category 
is the replete image of the fully faithful functor 
. A dTT natural model is a dTT context structure that is also a modal natural model.
Next, since our modalities are Fitch-style, their semantics follows Gratzer et al. (Reference Gratzer, Cavallo, Kavvos, Guatto and Birkedal2022). This requires each functor 
to be a parametric right adjoint and have a dependent right adjoint. However, since each 
other than 
and 
is already a right adjoint in 
, such 
are also ordinary (hence parametric) right adjoints. And since 
is an equivalence onto a slice category, the inverse of that equivalence is a parametric left adjoint of it. Thus, to justify our Fitch-style rules for modalities, it suffices to assume the following (recall we omit 
).
Definition 20. A dTT modal model is a dTT natural model such that the functors 
and 
have dependent right adjoints.
Proof. Of course, we use the extended simplicial model along with the discrete model. We showed explicitly in section 4.3 that this yields a modal context structure for our 
. In the notation of that section, the object 
referred to above is 
. The definition of the category 
implies immediately that this object is subterminal and its slice category is the replete image (and even the literal image) of 
. Finally, in section 4.3.5 we verified the rules of Gratzer et al. (Reference Gratzer, Cavallo, Kavvos, Guatto and Birkedal2022) for 
and 
, which as shown in loc. cit. are equivalent to their being dependent right adjoints of 
and 
.
4.4.2 Telescopes
Definition 22. A modal natural model 
has telescopes if each natural model 
has telescopes as in definition 9, and in addition for each nonidentity 
there are morphisms of polynomial functors 
(the identity case being part of 
having telescopes), satisfying the modal versions of the rules from section 2.3. It has 
-telescopes if 
does for each 
.
As in Sections 4.1.3 and 4.1.4, we can equip any modal natural model with telescopes and 
-telescopes, and interpret meta-abstracted types and telescopes automatically.
4.4.3 Display and décalage
If 
is a CwF with telescopes and 
, we write 
for the category whose objects are telescopes 
and whose morphisms are morphisms 
in 
. Thus, it is equivalent to the full subcategory of 
on objects of the form 
. We call this the category of telescopes of 
. By substitution, it is strictly functorial in 
, i.e. we have a functor 
. Thus can regard it as an internal category in 
.
In fact, 
is itself a CwF. Its ‘types’ in ‘context’ 
are meta-abstracted telescopes 
. These are equivalent (but not equal) to telescopes in an extended context, i.e. the elements of 
. Similarly, meta-abstracted partial substitutions 
are equivalent to terms 
, and semantically to sections of such a projection, 
over 
. Comprehension is by telescope concatenation. Because 
, the CwF 
has the following special property.
Definition 23. A CwF 
is strongly democratic if every context is the comprehension of a unique type in the empty context.
Since all of the structure of 
is strictly stable under substitution in 
, these categories form an internal CwF in 
. We call this the internal telescope model of 
and denote it 
. Recalling remark 5 and comparing to the discussion of the local theory in Altenkirch et al. (Reference Altenkirch, Chamoun, Kaposi and Shulman2024), we find that décalage can be described as an internal CwF morphism.
Definition 24. Let 
be a dTT natural model with telescopes. We say it has décalage if it is equipped with a strict morphism of internal strongly democratic CwFs:
in the category of presheaves over 
, together with an internal natural transformation 
from 
to 
.
In particular, this structure includes functors 
and natural transformations 
consisting of maps 
all strictly stable under pullback. This, together with the strict preservation of empty contexts by the functor 
, yields the rules of section 2.3. The corresponding action on ‘types’ (meta-abstracted telescopes) and ‘terms’ (meta-abstracted partial substitutions) gives the rules of section 2.5.
We can phrase display similarly with an auxiliary internal CwF, starting with telescope display as in section 2.5. Recall from Shulman (Reference Shulman2015), Kapulkin and Lumsdaine (Reference Kapulkin and Lumsdaine2021) that from any CwF 
we can construct a ‘Sierpinski’ model 
. Its objects (contexts) are arbitrary morphisms 
in 
, but its types are pairs of 
and 
. Moreover, there is a strict CwF morphism 
which preserves all type-formers. If 
has 
-types there is a functor 
that sends a type 
to 
, but this only preserves comprehension and 
-types up to isomorphism. Thus it is a pseudo CwF morphism as in (Clairambault and Dybjer, Reference Clairambault and Dybjer2011, Definition 10).
Now let 
be a CwF, and apply this construction internally in 
to the internal telescope model 
. We thus obtain another CwF 
internal to presheaves, in which the ‘contexts’ over 
are morphisms of telescopes 
over 
, and the ‘types’ in such a ‘context’ over 
are pairs of two telescopes 
and 
. This is no longer strongly democratic, but as always we have the strict CwF morphism 
that preserves 
-types (i.e. telescope concatenation), and the pseudo CwF-morphism 
. Moreover, in this case the latter is actually a strict CwF-morphism, because telescope concatenation is strictly associative.
Definition 25. Let 
be a dTT natural model with telescopes. We say 
has telescope display if it is equipped with:
-
1. An internal pseudo CwF morphism
that preserves substitution, the empty context, and -types strictly. -
2. An equality between the composite
and the key transformation . Since the latter is a strict morphism, that means that so is the former. -
3. A strict internal CwF morphism
, and an isomorphism of pseudo CwF morphisms between and the composite morphism , that is the identity on underlying functors.
that preserves substitution, the empty context, and 
-types strictly.
and the key transformation 
. Since the latter is a strict morphism, that means that so is the former.
, and an isomorphism of pseudo CwF morphisms between 
and the composite morphism 
, that is the identity on underlying functors. We have not assumed a priori in definition 25 that 
has décalage, but it is actually included: the morphism 
is of course the same as in definition 24, and the transformation 
from definition 24 arises in definition 25 as the image of 
under the underlying functor of 
. The additional data in definition 25 beyond this is the 
-part of the action of 
(the 
-part is determined by the composition with 
equaling 
) making it a pseudo CwF morphism preserving substitution and 
-types strictly, and the isomorphism on ‘types’ (dependent telescopes) between 
and the composite of 
with 
. But since the latter is to be a pseudo CwF transformation (see Castellan et al. Reference Castellan, Clairambault and Dybjer2017, Appendix B), and since 
is strict and the underlying functor is the identity this just means that this isomorphism must coincide with the 
-part of the comprehension coherence isomorphism of 
(the 
-part being the identity).
So all that remains is the 
-part of the action of 
on meta-abstracted telescopes, preserving substitution and telescope concatenation, and coherence isomorphisms relating it to comprehension. Since the 
-part of the comprehension of 
in 
is 
, the comprehension isomorphisms are the pairing 
together 
and 
.
Finally, we consider display of types. In some ways this is simpler, since we don’t have to worry about rearranging between display and décalage; but in other ways it is more complicated, since we have to take account of extending dependent telescopes by types.
To start with, note that the internal telescope model 
of any CwF has a ‘sub-model’ 
whose internal category of ‘contexts’ is the same (telescopes), but whose internal presheaf of ‘types’ consists of the length-
telescopes, i.e. single types annotated by a modality. Unlike 
, it does not automatically have 
-types.
Definition 26. Let 
be a dTT natural model with telescopes. We say 
has type display if it is equipped with:
-
1. An internal strict CwF morphism
. -
2. An equality between the composite
and the key transformation . -
3. If a length-1 telescope
is non-modal, then is a single non-modal type. -
4. If a length-1 telescope
is nontrivially modal, then is empty.
.
and the key transformation 
.
is non-modal, then 
is a single non-modal type.
is nontrivially modal, then 
is empty.Note that, like definition 25, this definition includes décalage. Of course, when we have both telescope display and type display we want them to be compatible.
Definition 27. Let 
be a dTT natural model with telescope display. We say it has complete display if the restriction of 
to 
is an internal strict CwF morphism such that items 3 and 4 of definition 26 hold, as do the rules from sections 2.4 and 2.5 for computing meta-abstracted décalage and display in terms of type display.
Finally, we add the compatibility conditions with type-formers:
Definition 28. Let 
be a dTT natural model with telescopes and complete display. We say that display respects 
-types (respectively universes) if the relevant rules in Figure 8
This completes the description of the abstract categorical semantics of the theory of section 2: it is a dTT natural model with telescopes and complete display that respects 
-types and universes. However, as noted in section 2, when telescopes are lists of types, as they almost always are, much of this structure can be deduced from the rest.
Theorem 29. Let 
be a dTT natural model, with telescopes defined from types as in theorem 10, and with type display defined relative to these telescopes. Then there is a unique way to extend this type display on 
to complete display.
Proof. The rules for computing telescope display and décalage in terms of type display uniquely determine those operations when telescopes are defined as lists of types.
Theorem 30. The simplicial model of Sections 4.2 and 4.3 has type display, and hence complete display, which respects 
-types and universes.
Proof. The display operation that we constructed for the simplicial model in section 4.2, is a ‘global’ operation that décalages the whole context:
(Note that this is only defined on the original simplicial model 
, not the extended one 
: indeed, décalage is not even defined on flat contexts.) But the (meta-abstracted version of the) operation we specified in the syntax of section 2 is a ‘local’ one that only décalages part of the context, keeping the rest of it modally locked away:
However, it is straightforward to obtain the latter from the former. In 
, a context of the form 
is not flat, hence lies essentially in 
so that décalage is defined on it. Furthermore, we already observed that 
since 
lands in constant presheaves. Thus, when 
is a telescope built out of types, we have:
and so the global operation yields as a special case:
Now we simply substitute along 
to obtain the desired local rule. The necessary computation rules for décalage, 
-types, and universes follow immediately.
4.4.4 Display of -limits
-limits Finally, when we have both display and also 
-limits, it is reasonable to require the former to compute on the latter, in the following way. Suppose that 
, and we want to compute 
. Then by definition, we have:
and therefore:
Weakening and substituting to the needed context 
, we have:
such that:
Thus, these data form another infinite telescope, which we denote:
Definition 31. In a CwF with telescopes, type display, and 
-limits, we say that display respects 
-limits if:
where in the last two equations, the left-hand side is a restriction relative to 
, and on the right-hand side it is relative to 
.
Theorem 32. Display respects 
-limits in the simplicial model.
Proof. This holds essentially by construction of 
-limits therein, plus passing across the translation between different forms of display from 30.
4.5 Semantics of semi-simplicial types
Finally, we construct semantics for the displayed coinductive types of section 3.3, including 
. As with most coinductive definitions, they are terminal coalgebras – in this case, for a copointed endofunctor – and can be constructed by a sequential limit.
4.5.1 Terminal coalgebras for copointed endofunctors
Definition 33. A copointed endofunctor of a category 
is a functor 
together with a natural transformation 
. A coalgebra for a copointed endofunctor is an object 
with a morphism 
such that the composite 
is the identity. A terminal coalgebra is a terminal object of the category of coalgebras.
To construct copointed terminal coalgebras, we cannot simply consider the limit of the tower 
as in the unpointed case; we have to incorporate 
.
Definition 34. (c.f. [Shulman Reference Shulman2019, Definition 8.6]). Given a natural transformation 
and a morphism 
in the domain of 
and 
, we write 
for the gap map in the following pullback, assuming that the pullback exists:
If the domain and codomain of 
and 
have a notion of ‘fibration’, we say that 
is a Quillen pre-fibration if whenever 
is a fibration, so is 
.
Lemma 35. In a category of telescopes, consider the fibrations to be the morphisms isomorphic to a dependent projection of some telescope. Then the transformation 
from definition 24 is a Quillen pre-fibration.
Proof. Given a dependent projection 
in 
, the gap map is isomorphic to the dependent projection 
of 
.
Theorem 36. Suppose 
is a category with a terminal object and a notion of fibration that is stable under pullback, and that 
is a copointed endofunctor of 
such that 
is a Quillen pre-fibration. Suppose also that 
has limits of inverse 
-sequences of fibrations, and that 
preserves these limits. Then there is a terminal 
-coalgebra.
Proof. We will construct a sequence of objects 
, with fibrations 
and morphisms 
, such that 
and 
. Let 
, the terminal object, and let 
, with 
the identity.
Now, assume the data constructed up to level 
; we want 
to be the universal object equipped with 
and 
satisfying the desired equations. The usual approach (e.g. the dual of Kelly Reference Kelly1980) is to take the equaliser:
but this does not make it evident that 
is a fibration. Instead, we take the pullback:
The commutativity of this square says that 
and 
. And by assumption, 
is a fibration, hence so is its pullback 
.
Now let 
be the limit of the 
-sequence of fibrations:
Since 
preserves 
-limits, 
is the limit of the corresponding sequence:
The morphisms 
and 
form fence diagrams:
composed of the parallelograms 
from our construction, and naturality squares 
. The former induces a map of limits 
, while by naturality the latter induces 
. The universal property of limits implies that 
is induced by the composite fence, and since 
this is:
which induces the identity 
. Thus, 
is an 
-coalgebra.
Now suppose 
is another 
-coalgebra. We construct inductively maps 
such that 
and 
. We start with 
the unique morphism, and 
the composite 
. Then we induce 
by the universal property of the pullback defining 
:
This is valid because using the inductive assumptions about 
and 
, we have:
and the two triangles relating to 
show that it has the necessary properties.
Now, the equations 
imply there is an induced map 
, such that 
is induced by the composites 
. But 
, and the morphisms 
induce the limit map 
. Thus, 
is an 
-coalgebra morphism.
Finally, suppose 
is any 
-coalgebra morphism, so we have 
. Then 
is uniquely determined by the maps 
, and we have 
. But this equation implies by induction that 
for all 
, hence 
.
4.5.2 Displayed coinductive types
Let 
be a dTT natural model with all the structure from section 4.4. We will apply theorem 36 in 
, in which the fibrations are the morphisms isomorphic to the dependent projection from some telescope. In the presence of levels, the objects of 
are telescopes at any level.
Suppose given the input data for a displayed coinductive type, consisting of:
Categorically, this yields a ‘display polynomial’. Omitting the base context 
for conciseness and indicating fibrations with 
, this looks like:
Here the left vertical map is a fibration because it is isomorphic to the dependent projection: 
. We then have a copointed endofunctor of 
:
Here the 
denotes a 
-telescope and 
denotes meta-abstracted telescope display. Note that 
is meta-abstracted over 
extended by 
, so it lies in 
rather than 
. The actual endofunctor of 
is thus
Lemma 38. The projection 
defines a Quillen pre-fibration 
.
Proof. Suppose given a fibration over 
, meaning a dependent telescope:
Then we have:
which we write as 
for conciseness. Then by definition, we have:
To simplify this, note that the computation rules for telescope display give
and therefore the rules for computing 
-telescopes give
Now when 
is paired with 
, it yields 
. Thus, the gap map 
is the dependent projection from the telescope:
and thus a fibration.
Lemma 40. The endofunctor 
preserves inverse limits of 
-sequences of fibrations.
Sketch of proof. This follows from the 
-rules for inverse limits, together with the fact that display also preserves inverse limits.
Therefore, by theorem 36, there is a terminal 
-coalgebra, which is a type (since 
-limits are types). This is our candidate for the displayed coinductive type. Its construction produces a tower of fibrations 
, i.e. a sequence of finite telescopes:
We choose the level of 
to be 
; then the other 
and 
are also at level 
.
The object 
in section 4.5.1 corresponds to the telescope:
Each morphism 
such that 
then corresponds to a term:
By definition of 
, 
is equivalent to two terms:
The equation 
means that:
for some term:
Inspecting the actual construction, we start with 
and 
. By induction, the functions 
then all just project to 
. For the rest, combining eqs. (37) and (39), we find that 
is defined by:
and we have 
. In particular, each 
is just a single type. Therefore, 
is an infinite telescope and our displayed coinductive type is its limit:
Example 41. Recall that for semi-simplicial types 
, we have 
, with 
and 
. Therefore, in this case we have:
This suggests that in general, 
will be the type of 
-truncated semi-simplicial types, while 
will be the type of ways to extend such an 
to an 
-truncated one, i.e. the types of indexed families of 
-simplices. We will prove this formally in section 4.5.5.
It remains to give 
the structure from section 3.3. A type 
is an 
-coalgebra when it has a section of the projection 
, that is:
This is equivalent to giving its components, which we abstract over 
:
This is the structure of 
and 
from section 3.3; thus 
admits these. Furthermore, to give any 
an 
-coalgebra structure is equivalent to giving 
the premises of the corecursor, where 
is the dependent projection. In the next section we show that for more general 
, the premises of the corecursor still equip 
with ‘enough of an 
-coalgebra structure’ to model the corecursor.
4.5.3 Terminal generalised coalgebras
Let 
be a copointed endofunctor of a category 
, where 
is a full subcategory.
Definition 42. An object 
is a generalised 
-coalgebra if it is equipped with, for any 
and 
, a specified morphism 
such that 
and for any 
in 
, we have 
.
More abstractly, 
induces a copointed endofunctor 
of 
by precomposition, and 
is a generalised 
-coalgebra if the functor 
is an 
-coalgebra.
Lemma 43. If 
, then generalised 
-coalgebra structures on 
are bijective to ordinary 
-coalgebra structures.
Proof. This follows immediately from the above observation and the Yoneda lemma.
Of course, a morphism of generalised 
-coalgebras is a morphism 
such that for any 
we have 
.
Lemma 44. If 
is a generalised 
-coalgebra and 
is an 
-coalgebra in 
, then a morphism 
is a generalised 
-coalgebra morphism if and only if 
.
Proof. If it is a generalised 
-coalgebra map, then taking 
in 
we get 
. On the other hand, if 
then for any 
in 
we have 
, as desired.
Theorem 45. Let 
and 
be as in theorem 36, and let 
be a full subcategory of 
such that the embedding preserves the terminal object and the inverse limits of 
-sequences of fibrations. Then the terminal 
-coalgebra constructed in theorem 36 is also a terminal generalised 
-coalgebra.
Proof. Indeed, the proof of theorem 36 really only uses the generalised 
-coalgebra structure. Specifically, let 
be a generalised 
-coalgebra. We construct inductively maps 
such that 
and 
. We start with 
the unique morphism (since 
is also terminal in 
), and 
. Then we induce 
by the universal property of the pullback defining 
:
This is valid because using the inductive assumptions about 
and 
and the properties of generalised coalgebras, we have: 
and 
, and the two triangles relating to 
show that it has the necessary properties.
Now 
yields an induced map 
, such that 
is induced by 
. But 
, and the morphisms 
induce the limit map 
, so 
. Thus, by lemma 44, 
is an 
-coalgebra morphism.
Finally, suppose 
is any 
-coalgebra morphism, so we have 
. Then 
is uniquely determined by the maps 
, and we have 
. But this equation implies by induction that 
for all 
, hence 
.
4.5.4 The general corecursor
Suppose 
has the structure of the premises of the corecursor from section 3.3:
Then 
makes it an object of the slice category 
. We will apply theorem 45 to the full subcategory 
. To that end, we give 
the structure of a generalised 
-coalgebra as follows. Suppose 
, and suppose we have a map 
, which is to say:
We want to lift 
to 
, which is to say we want to give:
But such an 
is exactly part of the structure of 
, while we can define:
The final equation in the structure of 
makes this well-typed, and functoriality is immediate from that 
. Thus 
is a generalised 
-coalgebra, hence admits a unique generalised 
-coalgebra morphism to 
. This is a generalized 
-coalgebra map 
over 
, which is precisely the right type of 
, and lemma 44 gives it the correct computation rules.
4.5.5 Correctness of semi-simplicial types
Finally, we will show that when 
is constructed as in section 4.5.2, it yields a ‘classifier’ of Reedy fibrant semi-simplicial types in the classical sense. We begin by constructing such a classifier category-theoretically, and then show that this construction coincides with the one in section 4.5.2. We will assume some familiarity with the classical notions of Reedy fibrant diagrams as in Kapulkin and Lumsdaine (Reference Kapulkin and Lumsdaine2021). For all this section, we fix a particular level 
.
4.6 Ordered direct categories
Definition 46. A direct category is one such that the relation ‘there is a nonidentity arrow from 
to 
’ on its objects is well-founded. A sieve in a (direct) category is a full subcategory 
such that if 
and 
, then 
. An ordered direct category is a finite direct category together with (1) a total ordering on its objects such that if 
then 
, and (2) such that for all objects 
, the set of arrows with codomain 
has a linear order such that 
for any composable 
(hence in particular 
is the greatest element).
An ordered presheaf on a direct category is a finite presheaf together with a linear order on the finite set 
such that 
whenever the left-hand side makes sense.
An ordered direct category is the opposite of a (finite) ‘ordered inverse category’ as in (Kapulkin and Lumsdaine, Reference Kapulkin and Lumsdaine2021, Definition 3.17), plus the object ordering (to order the variables in the classifying context). An ordered presheaf is a ‘finite extension’ 
as in (Kapulkin and Lumsdaine, Reference Kapulkin and Lumsdaine2021, Definition 3.10).
Example 47. Let 
be the subcategory of 
containing the objects 
with 
. For fixed 
we give these morphisms Campion’s ordering, namely the usual ordering of binary numbers. Then 
is an ordered direct category.
For 
we write 
for the sub-presheaf of the representable 
consisting of nonidentity morphisms, i.e. 
.
If 
is a finite direct category and 
is a finite presheaf on it, there is a new finite direct category 
, called the collage of 
, which contains 
as a full subcategory, together with one new object 
such that 
for all 
. Note that 
restricted to 
coincides with 
. Moreover, 
and 
are ordered if and only if 
is. Moreover, if 
is an ordered direct category of finite height with 
its object of greatest rank, then 
. Thus, we can treat this as an induction principle for ordered direct categories.
4.7 Classifying contexts
As our first use of this sort of induction, we construct by simultaneous induction:
-
(1) For each ordered direct category
, a context . This will be the classifying context of Reedy fibrant -types at level . -
(2) For each ordered presheaf
on , a telescope . -
(3) For each map of ordered presheaves
(not necessarily order-preserving) on , a partial substitution , varying functorially. -
(4) For each object
, a type . -
(5) For each
, inducing by the Yoneda lemma a map , a term , such that for any . -
(6) For each sieve
, a telescope such that and for all the structure in 2–5, the action of the weakening substitution corresponds to left Kan extension along the inclusion .
, a context 
. This will be the classifying context of Reedy fibrant 
-types at level 
.
on 
, a telescope 
.
(not necessarily order-preserving) on 
, a partial substitution 
, varying functorially.
, a type 
.
, inducing by the Yoneda lemma a map 
, a term 
, such that 
for any 
.
, a telescope 
such that 
and for all the structure in 2–5, the action of the weakening substitution 
corresponds to left Kan extension along the inclusion 
.For item 1, we inductively use item 2 and set:
For item 2, we argue inductively on the linear ordering of 
. If 
is empty, we set:
Otherwise, 
where 
is the last element in the ordering; the condition on the ordering ensures that 
is still an (ordered) presheaf. By the Yoneda lemma, 
induces a map 
, hence by item 3 a substitution 
. Thus, inductively using item 4 as well, we can define:
We similarly construct item 3 by induction on 
(the domain of 
). The case when 
is empty is trivial. Otherwise, we inductively have 
, and to extend the codomain to 
it suffices to give a term in context 
of type:
For this we can pick 
, using item 5 inductively. Functoriality follows inductively.
For item 4, note that the slice category 
is a sieve in 
containing 
. Then it suffices to define 
in the case of 
, since it can then be weakened to 
using item 6. In this case we have 
, so the last variable in 
is 
. Thus, we can define 
to be 
.
For item 5, it suffices to deal with the case when 
is the last element in the ordering of 
, since otherwise we can weaken from the sub-presheaf of all elements 
to all of 
, using item 3 for the inclusion of this sub-presheaf. But in this case, the last variable in 
is 
, so we can take 
. Functoriality follows immediately, as does stability under weakening from initial segments for all the data.
For item 6 we induct on 
. A sieve in 
could be either 
or 
for some sieve 
in 
, depending on whether it contains 
. (If it contains 
, it must contain all objects 
such that 
, so 
must be left Kan extended from 
.) In these two cases, we define:
This completes the construction. In particular, item 3 implies that re-ordering the elements of a presheaf 
modifies 
only up to isomorphism.
4.8 The classifying context is classifying
We first construct a ‘universal’ diagram over 
. Specifically, in any CwF, we construct simultaneously:
-
1. For each ordered direct category
, a Reedy type of shape and level over in the sense of (Kapulkin and Lumsdaine, Reference Kapulkin and Lumsdaine2021, Definition 3.22). -
2. For each ordered presheaf
on , the object is the -weighted limit of constructed by (Kapulkin and Lumsdaine, Reference Kapulkin and Lumsdaine2021, Lemma 3.11). In particular, is the matching object . -
3. The maps
are the functorial action of these limits. -
4. The type
is the object with its fibration to . -
5. The elements
are the projections from the weighted limit .
, a Reedy type 
of shape 
and level 
over 
in the sense of (Kapulkin and Lumsdaine, 
on 
, the object 
is the 
-weighted limit of 
constructed by (Kapulkin and Lumsdaine, 
is the matching object 
.
are the functorial action of these limits.
is the object 
with its fibration to 
.
are the projections from the weighted limit 
. The interesting case is item 1, where we weaken the Reedy 
-type 
over 
to 
and then must extend it to a Reedy 
-type by giving a type over the matching object 
. But 
(weakened to 
), and so we can use 
where 
is the newly added variable in 
. The other parts follow essentially tautologically.
Lastly, suppose 
is a Reedy 
-type over any context 
, and suppose that it is ‘
-small’ in the sense that each type 
over the matching object 
is classified by a specified map into the universe 
. We show by induction on 
that it is classified by a unique map 
such that 
. This is trivial when 
is empty. Assuming it to be true for 
, if 
is a Reedy 
-type over 
, and its restriction to an 
-type is classified by a map 
, then to extend this to a map into 
we must give a term in context 
of type 
. But substitution preserves matching objects, so 
is the matching object 
, and thus this is exactly the data extending 
to a Reedy 
-type.
4.9 Display and décalage of classifying contexts
Since 
, 
, 
, and so on are concrete syntactic objects (for any fixed 
and so on), the computation rules completely determine display and décalage on them. We can characterise the results as follows.
Let 
denote the interval category 
. Then if 
is a direct category, so is 
where 
yields 
. If 
is ordered, we order 
as follows. The morphisms in 
with codomain 
are bijective to those in 
with codomain 
, so we inherit that ordering. And the morphisms in 
with codomain 
are two copies of those in 
with codomain 
indexed by 
and 
respectively, so we order them where 
yields 
.
The projection 
induces by precomposition from any presheaf 
on 
a presheaf 
on 
. Each element 
then induces two elements 
in 
and 
; we denote these 
and 
respectively for clarity. If 
is ordered, we induce an ordering on 
by ordering each element of 
before the corresponding element of 
(which is necessary, since 
maps the latter to the former).
The inclusion 
defined by 
is a sieve. Left Kan extending along this inclusion takes a presheaf 
on 
to a presheaf 
on 
that is supported on objects 
and has exactly the same elements, hence inherits an ordering as well.
Now we prove by simultaneous induction:
-
(1) For any
, we have . -
(2) For any presheaf
on , and . -
(3) This identification is functorial in maps
. -
(4) For each
, we have:
-
(5) For each
, we have:
, we have 
.
on 
, 
and 
.
.
, we have:
, we have:
For the inductive step of item 1, we have:
Thus, using item 2, we have:
The other cases are similar. We can likewise show 
, the isomorphism 
coinciding with the evens/odds pairing 
.
4.10 Discrete fibrations
The isomorphism 
ensures that if 
is a sieve, we have a weakening substitution 
. But more generally, we can expect to induce a context substitution from any discrete fibration. Even more generally, we can get a partial substitution from a ‘dependent’ discrete fibration, in the following sense.
Definition 48. If 
is the inclusion of a sieve in a direct category, a co-section of it is a discrete fibration 
such that 
. In this case, if 
is a presheaf on 
and 
a presheaf on 
, a morphism 
over 
is a relative isomorphism if it induces a bijection 
.
The prototypical relative isomorphism is 
for any 
. Now we define and prove inductively:
-
1. For any co-section
of a sieve in an ordered direct category, a partial substitution . -
2. For any order-preserving relative isomorphism
, we have . -
3. For
, we have . -
4. For
, and an order-preserving relative iso, .
of a sieve 
in an ordered direct category, a partial substitution 
.
, we have 
.
, we have 
.
, and 
an order-preserving relative iso, 
. For item 1, as before a sieve in 
can be either 
or 
for a sieve 
in 
. In the latter case, we have 
weakened to 
, and a co-section of 
is determined by a co-section 
of 
; thus we can similarly weaken 
. In the former case, a co-section 
is determined by a co-section 
with 
and a relative isomorphism 
. Since 
, to extend 
as desired it suffices to give a term of type 
. But using item 2 inductively, this is equal to 
, so we can use the variable 
in 
.
We can then prove items 2–4 by inducting on the ordering of 
and 
.
4.11 Categorical coning
Let 
be a sieve in a direct category that contains the bottom object, which we denote 
. Let 
be 
augmented by a new morphism 
for all objects 
. (The notation is somewhat abusive, since the construction depends on 
as well as 
.) We define 
for all 
; note that 
implies 
since 
is a sieve. If 
is ordered, we order 
by placing 
before all other morphisms with codomain 
. Note that 
is still a sieve in 
. Similarly, for a presheaf 
on 
, let 
denote the presheaf on 
consisting of 
plus by a new element 
, such that 
for all 
. If 
is ordered, we order 
by putting 
first.
e now inductively prove:
-
(1) For any sieve
in an ordered direct category, we have (meaning a -telescope). -
(2) In addition, for any
on , if we transfer and across the isomorphisms:
in an ordered direct category, we have 
(meaning a 
-telescope).
on 
, if we transfer 
and 
across the isomorphisms:
to get 
and 
, then we have: 
.
Both are straightforward, using the inductive definition of 
-telescopes, 
, and 
.
4.12 Correctness of semi-simplicial types
Recall that our definition of semi-simplicial types 
is as a displayed coinductive type with 
, 
, and 
. Therefore, the construction in section 4.5.2 simplifies as follows:
We will prove inductively that:
This will imply that 
is a classifying context for all of 
. The claim about 
clearly inductively implies the claim about 
. Also it is easy to show inductively that 
. So it remains to say something useful about 
.
Let 
be the subcategory of 
containing all the objects except 
, and let 
regarded as a sieve in 
. The central fact is the following.
Lemma 49. For any 
, there is a co-section 
.
Proof. On objects, let 
for 
. A morphism 
is a length 
sequence with 
s, and we augment it by another 
on the right to get a length 
sequence with 
s, hence a morphism 
. A morphism 
is also a length 
sequence with 
s, but this time we augment it by a 
on the right to get a length 
sequence with 
s, hence a morphism 
. Finally, we send the new morphism 
to the sequence of 
s followed by one 
. Functoriality is easy to check. And to see that it is a discrete fibration, we observe that any binary sequence of length 
with a positive number of 
s must be of exactly one of these three forms: a positive number of 
s followed by a 
, a positive number of 
s followed by a 
, or a sequence of 
s followed by a 
.
Evidently 
restricts to 
as we shrink the categories. Thus, we also get a relative isomorphism 
over 
. Now note that if we abstract over 
, the type of 
matches that of 
. Thus, we can now prove by simultaneous induction that:
-
(1)
. -
(2)
. -
(3)
. -
(4)
.
.
.
.
.We have already remarked that items 1 and 3 are easy, and the base cases of items 2 and 4 are likewise trivial. For the induction step of item 2, we have:
Finally, the induction step of item 4 follows from the definition of 
and the inductive hypothesis of item 2. This completes the proof of the correctness of 
.
5. Conclusion and Future Work
In this paper we have made two main contributions. First, we have described Displayed Type Theory (dTT), a new kind of type theory that incorporates (unary) internal parametricity but guarded by a modality, and showed that any model of dependent type theory with countable Reedy limits can be lifted to a model of dTT using augmented semi-simplicial diagrams. Because the latter are diagrams on an inverse category, their type theory is more closely related to that of the original model, and indeed the original model sits inside our model of dTT at the discrete mode. In particular, unlike other internally parametric type theories, dTT is compatible with classical axioms such as excluded middle and choice, as long as they are formulated at the discrete mode (or under the modality 
), and can be used as an internal logic to reason about arbitrary
$(\infty,1)$
-toposes.
Secondly, inside dTT we have introduced a notion of displayed coinductive type, where the output of a destructor can be a parametricity ‘computability witness’ of the input, and showed that as a particular case of this notion we can define a type of semi-simplicial types. This yields a new approach to the long-standing open problem of representing infinitely coherent higher structures in type theory. Relative to other approaches, ours has the advantage that semi-simplicial types are defined (not postulated) as a simple instance of a type-former with natural introduction and elimination rules, i.e. a categorical universal property. While it remains to be seen how much can actually be done in practice with our definition, early indications of its utility are promising.
There are a number of directions for future work suggested by our results.
5.1 Computation and implementation
We conjecture that dTT, including the type 
of section 3.1 and the displayed coinductive types of section 3.3, satisfies canonicity, normalisation, and decidable typechecking. Indeed, while we have presented dTT only as a Generalised Algebraic Theory, all the equations are directed and there are no obvious stuck terms. Therefore, it should be possible to implement a proof assistant for dTT.
Some evidence in this direction is that, as mentioned in section 3, the second author has implemented a prototype proof assistant called Narya with a mode that makes its type theory similar to dTT. The major differences are that there are no modalities (instead, typechecking display clears the entire context), there are symmetries, and the computation rules for display only hold up to definitional isomorphism. Although Narya includes a normalisation algorithm, this algorithm is not yet proved correct; but the related proof of canonicity in Altenkirch et al. (Reference Altenkirch, Chamoun, Kaposi and Shulman2024) provides additional positive evidence for our conjecture.
5.2 Modal internal parametricity
We expect that most applications of ordinary internal parametricity have modal versions that can be proven in dTT (or a higher-ary version of it, as discussed below). In addition to the traditional ‘free theorems’, it would be interesting to investigate this for the proof of the pentagon identity for the smash product in Cavallo (Reference Cavallo2021), since such a proof would then apply internally to any 
-topos.
5.3 Computing diamond
Our intended semantic model strictly validates computation rules for diamond, such as 
. The syntax of dTT could be augmented with similar computation rules for diamond, by extending the definition of diamond to telescopes and meta-abstractions to handle open terms under binders.
5.4 Higher category theory
We have defined a type of semi-simplicial types in dTT, but such a definition is not an end in itself; it is intended as a tool for developing a theory of higher categories and other higher structures. We hope that our corecursion principle of 
, and the availability of other displayed coinductive types for predicates and structures on them, should make the development of such a theory feasible in dTT. We sketched some initial ideas in sections 3.2 and 3.4, but much remains to be done.
5.5 Elementary models
Our model construction in section 4 using 
-limits is probably not the only way to construct models of dTT. We conjecture there are ‘realisability’ models of dTT in which 
is a classifier for ‘uniform’ semi-simplicial types. If so, displayed coinductive types might be useful to include in a definition of elementary 
-topos.
5.6 Higher-ary dTT
The parametricity of dTT is unary, meaning 
depends on one copy of 
; but parametricity can be 
-ary for any natural number 
. We expect that higher-ary versions of dTT can be defined and modeled by a straightforward modification of this paper, using higher-ary semi-cubical types in place of augmented semi-simplicial types. In this case the binary numbers 
become base-
numbers.
5.7 Symmetries
In non-modal internal parametricity and Higher Observational Type Theory, it appears necessary to include a ‘symmetry’ operation, which in our notation would have type 
. The absence of symmetry in dTT is a significant simplification; e.g. it means that 
is a strict inverse category, allowing the explicit model construction in section 4.2. However, it also leads to certain limitations, e.g. without symmetry it is unclear how to give a corecursion principle for 
. It should be possible to add symmetries to dTT, but in the presence of symmetry it is unclear whether it is possible for display to compute definitionally on type-formers. However, it should work to use either the interval-based style of Bernardy and Moulin (Reference Bernardy and Moulin2012), Bernardy (2015), Moulin (Reference Moulin2016) or the ‘observational’ style of Altenkirch et al. (Reference Altenkirch, Chamoun, Kaposi and Shulman2024).
5.8 Unimode dTT
We have formulated dTT with two modes, but intuitively the discrete mode is unnecessary, as the 
-types are embedded in the 
-types by 
. Thus, it should be possible to formulate a single-moded version of dTT, similarly to spatial/cohesive type theory Shulman (Reference Shulman2018) and synthetic guarded domain theory Gratzer et al. (Reference Gratzer, Kavvos, Nuyts and Birkedal2021).
5.9 Conjectural syntax
One may also consider other kinds of generalised inductive and coinductive types, especially when taking a more ‘synthetic’ approach to higher structures in dTT, using 
-types directly as augmented semi-simplicial objects instead of
.
Firstly, regarding display as analogous to paths in homotopy type theory suggests displayed inductive types as analogues of higher inductive types, where the constructors generate displayed elements. As an example, we can construct the simplicial cone of any type:
Secondly, regarding both display and paths as modalities suggests more general modal inductive types, whose constructors can land in modal versions of the type. For instance, a 
-modal constructor adds a 
-simplex without any higher simplices above it. In this way we can construct the free-living simplices:
In both cases, we rely on the computation 
and 
to directly give induction principles. For example, the pattern match 
requires that 
of a function type compute to a function type. This is an improvement on the treatment of higher inductive types in Univalent Foundations Program (2013) using 
(a compound expression defined by path induction). In particular, we conjecture that in dTT, these displayed and modal inductive types can be made fully computational.
Acknowledgements.
Both authors would like to thank Tim Campion for bringing the binary ordering to their attention, via a talk given by Emily Riehl on her joint work with Tim. Astra would also like to thank Emily for for many discussions in the course of weekly advising meetings. Further, Astra is grateful to Steve Awodey for hosting her during the months of March and April 2023 at Carnegie Mellon University, and Emily for hosting her during the Fall 2023 semester at Johns Hopkins University. Many of the initial ideas regarding the semantics of dTT were developed during the CMU visit, and our construction of the simplicial model was worked out during the JHU visit. Mike is grateful to Thorsten Altenkirch, Ambrus Kaposi, and Yorgo Chamoun for many useful conversations while developing Higher Observational Type Theory that have also informed this work.
Astra: We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC). Cette recherche a été financée par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG). [funding reference number CGSD3-545891-2020]
This material is based upon work supported by the National Science Foundation under Grant No. DMS-2204304.
Mike: This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-21-1-0009.
-toposes have strict univalent universes. 
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