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 $U$).



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.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.