# Formalization and set-theoretic issues in the definition a functor category

Universes are used in a category theory to handle size-issues. Assuming Grothendieck's axiom UA that

every sets is contained in some universe

there are two approaches to $$U$$-smallness given a universe $$U$$.

1) A set is $$U$$-small if it is isomorphic to an element of $$U$$ (reference: SGA),

2) A set is $$U$$-small if it is an element of $$U$$ (references: Murfet, Low, Dwyer-Hirschhorn-Kan-Smith).

Of course, two main things that differs in these two approaches are definitions of a small $$U$$-category (resp., a $$U$$-small category) and of a $$U$$-category (resp., a locally $$U$$-small category).

In the second approach it's customary to say a $$U$$-category for a category whose set of objects is a subset of $$U$$ (some may want to drop this condition) and, more importantly, whose $$\mathsf{Hom}$$-sets are $$U$$-small with respect to 2), that is, are elements of $$U$$.

An important notion in category theory is a category of functors $$[\mathsf{C},\mathsf{D}]$$. The problem with the second approach to $$U$$-smallness is that $$[\mathsf{C},\mathsf{D}]$$ is not necessarily a $$U$$-category even if $$\mathsf{C}$$ is $$U$$-small and $$\mathsf{D}$$ is a $$U$$-category, unlike if we work with classes or with the first approach to $$U$$-smallness.

The definitions of a functor and a natural transformations that I use:

• A functor $$F\colon\mathsf{C}\to\mathsf{D}$$ is a special kind of quadruple $$(\mathsf{C},\mathsf{D},F_0\colon\mathsf{Ob(C)}\to\mathsf{Ob(D)}, F_1\colon\mathsf{Mor(C)}\to\mathsf{Mor(D)}),$$
• A natural transformation $$\alpha\colon F\Rightarrow G$$ is a special kind of triple $$(F,G,(\alpha_X\colon F(X)\to G(X))_{X \in \mathsf{Ob(C)}})$$.

These definitions are motivated by the assumption that every functor (resp, every natural transformation) needs to have a well-defined domain and a well-defined codomain. However, they get in the way of a category of functors between a $$U$$-small category and a $$U$$-category being a $$U$$-category. But what if we redefine a functor category in terms of sets?

Say, given categories $$\mathsf{C}$$ and $$\mathsf{D}$$, the functor category $$[\mathsf{C},\mathsf{D}]$$ is not a category of functors and of natural transformations per se, but a category

• whose objects are pairs $$(F_0,F_1)$$ of surjective functions which define a functor $$F\colon\mathsf{C}\to\mathsf{D}$$,

• whose morphisms between two such pairs $$(F_0,F_1)$$ and $$(G_0,G_1)$$ are triples $$((F_0,F_1),(G_0,G_1),A)$$ where $$A = \{ (X,\alpha_X) \mid X \in \mathsf{Ob(C)} \}$$ which defines a natural transformations between respective functors which are defined by $$(F_0,F_1)$$ and $$(G_0,G_1)$$

• the composition law and the identities are evident.

Of course, it is only a technicality, and given two categories $$\mathsf{C}$$ and $$\mathsf{D}$$ we can identify objects and morphisms of our newly defined $$[\mathsf{C},\mathsf{D}]$$ with "real" functors and natural transformations.

Zhen Lin Low seems to do something similar is his aforementioned notes, that is, he says that such a category is isomorphic to a category of functors and Daniel Murfet seems to also do it implicitly (or maybe he uses a different definition of functors and natural transformations).

My question is this: is redefining a functor category such as this safe? Wouldn't it lead to any problems with other constructions?

• What is your definition of function? – Fred Rohrer Nov 26 '18 at 15:51
• Why do you want the maps $F_0$ and $F_1$ to be surjective? – Fred Rohrer Nov 27 '18 at 8:03
• The codomain of $F_0$ or $F_1$ is the set of objects or morphisms of ${\sf D}$, which need be neither an element of $U$ nor $U$-small. But anyway, not every functor (in the usual sense) can be described by two surjective maps to the set of objects or morphisms of its target category. – Fred Rohrer Nov 27 '18 at 9:38
• @FredRohrer No, I don't request a set of objects to be an element of $U$ in a $U$-category, merely a subset of $U$. Hence $U$-$\mathsf{Set}$, $U$-$\mathsf{Grp}$ etc.are $U$-categories. For the proof why my $[\mathsf{C},\mathsf{D}]$ is a $U$-category you can see linked notes of Murfet or Low. As for considering surjective versions of $F_0\colon\mathsf{Ob(C)}\to\mathsf{Ob(D)}$ and $F_1\colon\mathsf{Mor(C)}\to\mathsf{Mor(D)}$ instead these functions themselves, you may instead consider relations $F_0$ and $F_1$ as sets of ordered pairs, it's essentially the same as – Jxt921 Nov 27 '18 at 10:26
• functional relations and surjective functions are in bijection. We only need to define functions as pairs $(f,Y)$ where $f$ is functional relations with $\mathrm{ran}(f) \subseteq Y$ because we want to consider nonsurjective functions. – Jxt921 Nov 27 '18 at 10:27