Pseudolimits for Tangent Categories with Applications to Equivariant Algebraic and Differential Geometry
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arXiv
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| Format: | Preprint |
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2023
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| author | Pronk, Dorette Vooys, Geoff |
| author_facet | Pronk, Dorette Vooys, Geoff |
| contents | In this paper we show that if $\mathscr{C}$ is a category and if $F\colon\mathscr{C}^{\operatorname{op}} \to \mathfrak{Cat}$ 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 $(F(f),{}_{f}α)$ and that furthermore the natural transformations ${}_{f}α$ vary pseudonaturally in $\mathscr{C}^{\operatorname{op}}$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_11753 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Pseudolimits for Tangent Categories with Applications to Equivariant Algebraic and Differential Geometry Pronk, Dorette Vooys, Geoff Category Theory Algebraic Geometry Primary 18F40, Secondary 18F20, 14A99, 53C10 In this paper we show that if $\mathscr{C}$ is a category and if $F\colon\mathscr{C}^{\operatorname{op}} \to \mathfrak{Cat}$ 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 $(F(f),{}_{f}α)$ and that furthermore the natural transformations ${}_{f}α$ vary pseudonaturally in $\mathscr{C}^{\operatorname{op}}$, 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. |
| title | Pseudolimits for Tangent Categories with Applications to Equivariant Algebraic and Differential Geometry |
| topic | Category Theory Algebraic Geometry Primary 18F40, Secondary 18F20, 14A99, 53C10 |
| url | https://arxiv.org/abs/2308.11753 |