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A set $U$ is a universe if

  • for any $x \in U$ we have $x \subseteq U$,

  • for any $x,y \in U$ we have $\{x,y\} \in U$,

  • for any $x \in U$ we have $\mathcal{P}(x) \in U$,

  • for any family $(x_i)_{i \in I}$ of elements $x_i$ of $U$ indexed by an element $I$ of $U$ we have $\bigcup_{i\in I} x_i \in U$

  • $\mathbb{N} \in U$.

Let $U$ be a universe. We say that a category $\mathcal{C}$ is a $U$-small if $Ob(\mathcal{C}) \in U$ and $Mor(\mathcal{C}) \in U.$ We say that a category $\mathcal{C}$ is a $U$-category if $Ob(\mathcal{C}) \subseteq U$ and each $Hom_{\mathcal{C}}(X,Y) \in U.$

Let $U$ be a universe, let $C$ be a $U$-small category and let $D$ be a $U$-category. Is it true that the category $[C,D]$ of functors between these categories is necessarily a $U$-category? The result is stated in SGA, but they use a slightly different definition of a $U$-category (in particular, the set of objects of their $U$-category doesn't have to be a subset of $U$).

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By SGA:

1). A set $X$ is called $\mathcal{U}$-small, if there exists a set $Y$, such that $Y\in\mathcal{U}$ and $X\cong Y$;

2). A category $\mathcal{C}$ is called $\mathcal{U}$-small, if the sets $\text{Obj}(\mathcal{C})$ and $\text{Mor}(\mathcal{C})$ are $\mathcal{U}$-small.

3). A category $\mathcal{C}$ is called a $\mathcal{U}$-category, if for every $c_1,c_2\in\text{Obj}(\mathcal{C})$ the set $\hom_{\mathcal{C}}(c_1,c_2)$ is $\mathcal{U}$-small.

Indeed, if $\mathcal{C}$ is a $\mathcal{U}$-small category and $\mathcal{D}$ is a $\mathcal{U}$-category, then the category of functors from $\mathcal{C}$ to $\mathcal{D}$ is a $\mathcal{U}$-category. The proof is that the set of natural transformations between $\mathcal{F}\colon\mathcal{C}\to\mathcal{D}$ and $\mathcal{G}\colon\mathcal{C}\to\mathcal{D}$ is isomorphic to a subset of $\prod_{c\in Obj(\mathcal{C})}\hom_{\mathcal{D}}(\mathcal{F}(c),\mathcal{G}(c))$, which is $\mathcal{U}$-small.

By your definitions:

1). A category $\mathcal{C}$ is $\mathcal{U}$-small, if $\text{Obj}(\mathcal{C})\in\mathcal{U}$ and $\text{Mor}(\mathcal{C})\in\mathcal{U}$;

2). A category $\mathcal{C}$ is a $\mathcal{U}$-category, if $\text{Obj}(\mathcal{C})\subset\mathcal{U}$ and $\hom_{\mathcal{C}}(c_1,c_2)\in\mathcal{U}$ for every $c_1,c_2\in\text{Obj}(\mathcal{C})$.

Of course, your definitions are not equivalent to those from SGA. Then the analogous statement about functor categories is wrong or requires very unnatural definitons to be true. For example, if a functor $\mathcal{F}\colon\mathcal{C}\to\mathcal{D}$ is a quadruplet $(\mathcal{C},\mathcal{D},\mathcal{F}_{\text{Obj}},\mathcal{F}_{\text{Mor}})$, then the category of functors $\mathcal{D}^{\mathcal{C}}$ fails to be a $\mathcal{U}$-category, because every functor should be an element of $\mathcal{U}$, but a quadruplet $(\mathcal{C},\mathcal{D},\mathcal{F}_{\text{Obj}},\mathcal{F}_{\text{Mor}})$ is not (simple set-theoretical formal verification). On the other hand, if a functor $\mathcal{F}\colon\mathcal{C}\to\mathcal{D}$ is a pair $(\mathcal{F}_{\text{Obj}},\mathcal{F}_{\text{Mor}})$ and a natural transformation $\alpha\colon\mathcal{F}\to\mathcal{G}$ is a subset of $\text{Obj}(\mathcal{C})\times\text{Mor}(\mathcal{D})$ (not a triplet $(\mathcal{F},\mathcal{G},R_{\alpha})$, where $R_{\alpha}\subset \text{Obj}(\mathcal{C})\times\text{Mor}(\mathcal{D})$), then there may happen a situation where domain and codomain of a natural transformation are not defined. And only if you adjust definitions of category, functor and natural transformation such that the set of natural transformations between $\mathcal{F}$ and $\mathcal{G}$ will be exactly a subset of $\prod_{c\in Obj(\mathcal{C})}\hom_{\mathcal{D}}(\mathcal{F}(c),\mathcal{G}(c))$, then this statement will be true. The reason why authors of SGA have chosen the definitions of $\mathcal{U}$-small category and $\mathcal{U}$-category with the requirements to corresponding sets only being isomorphic to elements of $\mathcal{U}$, is that, I guess, they didn't want to deal with these irrelevant set-theoretical issues.

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  • $\begingroup$ Dear Oskar, is there any way I could contact you outside this site? I have a few questions regarding universes. $\endgroup$ – Jxt921 Nov 23 '18 at 18:01
  • $\begingroup$ @Jxt921 Yes, you can contact me by e-mail: oskar808@mail.ru $\endgroup$ – Oskar Nov 23 '18 at 18:29

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