In this paper, we show that if $\mathscr{C}$ is a category and if $F\colon \mathscr{C}^{\;\textrm {op}} \to \mathfrak{Cat}$ is a pseudofunctor such that for each object $X$ of $\mathscr{C}$ the category $F(X)$ is a tangent category and for each morphism $f$ of $\mathscr{C}$ the functor $F(\,f)$ is part of a strong tangent morphism $(F(\,f),\!\,_{f}{\alpha })$ and that furthermore the natural transformations $\!\,_{f}{\alpha }$ vary pseudonaturally in $\mathscr{C}^{\;\textrm {op}}$, then there is a tangent structure on the pseudolimit $\mathbf{PC}(F)$ which is induced by the tangent structures on the categories $F(X)$ together with how they vary through the functors $F(\,f)$. We use this observation to show that the forgetful $2$-functor $\operatorname {Forget}:\mathfrak{Tan} \to \mathfrak{Cat}$ creates and preserves pseudolimits indexed by $1$-categories. As an application, this allows us to describe how equivariant descent interacts with the tangent structures on the category of smooth (real) manifolds and on various categories of (algebraic) varieties over a field.

The large-structure tools of cohomology including toposes and derived categories stay close to arithmetic in practice, yet published foundations for them go beyond ZFC in logical strength. We reduce the gap by founding all the theorems of Grothendieck’s SGA, plus derived categories, at the level of Finite-Order Arithmetic, far below ZFC. This is the weakest possible foundation for the large-structure tools because one elementary topos of sets with infinity is already this strong.

Given a $2$-commutative diagram

in a symmetric monoidal $(\infty ,2)$-category $\mathscr{E}$ where $X,Y\in \mathscr{E}$ are dualizable objects and $\unicode[STIX]{x1D711}$ admits a right adjoint we construct a natural morphism $\mathsf{Tr}_{\mathscr{E}}(F_{X})\xrightarrow[{}]{~~~~~}\mathsf{Tr}_{\mathscr{E}}(F_{Y})$between the traces of $F_{X}$ and $F_{Y}$, respectively. We then apply this formalism to the case when $\mathscr{E}$ is the $(\infty ,2)$-category of $k$-linear presentable categories which in combination of various calculations in the setting of derived algebraic geometry gives a categorical proof of the classical Atiyah–Bott formula (also known as the Holomorphic Lefschetz fixed point formula).

The notion of a prolongation of an algebraic variety is developed in an abstract setting that generalizes the difference and (Hasse) differential contexts. An interpolating map that compares these prolongation spaces with algebraic jet spaces is introduced and studied.

Let k be a complete, non-Archimedean valued field (the trivial absolute value is allowed) and let φ:X→Y be a morphism between two Berkovich k-analytic spaces; we show that, for any integer n, the set of points of X at which the local dimension of φ is at least equal to n is a Zariski-closed subset of X. In order to establish it, we first prove an analytic analogue of Zariski’s Main Theorem, and we also introduce, and study, the notion of an analytic system of parameters at a point.