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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?

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    $\begingroup$ What is your definition of function? $\endgroup$ – Fred Rohrer Nov 26 '18 at 15:51
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    $\begingroup$ Why do you want the maps $F_0$ and $F_1$ to be surjective? $\endgroup$ – Fred Rohrer Nov 27 '18 at 8:03
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    $\begingroup$ 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. $\endgroup$ – Fred Rohrer Nov 27 '18 at 9:38
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    $\begingroup$ @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 $\endgroup$ – Jxt921 Nov 27 '18 at 10:26
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    $\begingroup$ 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. $\endgroup$ – Jxt921 Nov 27 '18 at 10:27

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