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Pseudolimits for tangent categories with applications to equivariant algebraic and differential geometry
- Part of
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- Journal:
- Mathematical Structures in Computer Science / Volume 35 / 2025
- Published online by Cambridge University Press:
- 27 November 2025, e33
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- Article
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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.
