Set-theoretic issues regarding functor categories

[catdat-issue-creation]

If $C$ is small and $D$ is locally small, then I believe whether $[ C, D]$ is locally small depends on technicalities about how natural transformations and functors have been coded in set theory. It is certainly essentially locally small, but as far as I can tell it need not actually be locally small under the codings given here. The issue is that a natural transformation is defined a map with codomain $\mathrm{Ob}(D)$, which may be a collection (such as the collection of all sets).

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This issue has been created by Joe Lamond via the submission form on https://catdat.app/content/foundations

[JoeLamond6]

Whoops, I should have said that a natural transformation has codomain $\mathrm{Mor}(D)$, which may be a collection and not a set.

[ScriptRaccoon]

Thanks for your submission!

Let $F,G : C \rightrightarrows D$ be two functors, where $C$ is small (we need less: we only need that $Ob(C)$ is a set) and $D$ is locally small. Let $\alpha : F \to G$ be a morphism. This is a map $\alpha : Ob(C) \to Mor(D)$ with some properties. We don't need naturality, we only need that $\alpha(x)$ is a morphism from $F(x)$ to $G(x)$, i.e. it belongs to the set $Hom(F(x),G(x))$, for every $x \in Ob(C)$. Let $M$ be the union of the sets $Hom(F(x),G(x))$, where $x \in Ob(C)$. This is a set since it is a set-indexed union of sets. Then $\alpha$ is a map $Ob(C) \to M$. This shows that $Hom(F,G)$ is contained in the set $Hom(Ob(C),M)$. Hence, it is a set. Thus, $[C,D]$ is locally small.

Problems might arise if we insist that a natural transformation is not just a map, but rather a triple $(F,G,\alpha)$, so that we can define source and target. Or, if we insist that the map $\alpha$ itself is not just a set of ordered pairs, but rather a triple $(Ob(C),Mor(D),\Gamma_{\alpha})$. 🤔 In any case, these are just encoding problems, as you correctly point out.

This question can also be seen as a part of #83 (and #25). For example, it is argued there that we should change the definition of "locally small" to what currently is listed as "locally essentially small". Maybe you can comment over there as well. Unfortunately, #83 is not resolved yet (but the idea is clear).

[JoeLamond6]

There are certainly no serious issues here.

One natural way of coding functor categories so that the desired theorem literally holds is described in the paper Universes for Category Theory by Zhen Lin. (This paper uses different conventions regarding sizes compared to the foundations section in CatDat, but the coding described by Lin still works in this context.) Quoting from the paper: "In the context of [D, C], we may regard functors D → C as being the pair consisting of the graph of the object map obD → obC and the graph of the morphism map morD → morC, and these are U-sets by the U-replacement axiom. Similarly, if F and G are objects in [D,C], then we may regard a natural transformation α : F ⇒ G as being the triple (F,G,A), where A is the set of all pairs (c, αc)." Perhaps we could add a footnote explaining this in the foundations section, but it might be unnecessary.

I prefer the coding described by Lin, because we want the natural transformations to carry the data of F and G, in order for the arrows in the functor category to have an unambiguous source and target. On the other hand, we don't need the functors themselves to carry data about the categories C and D in this context. (And if they did carry such data, then annoying set-theoretic problems would arise.)

[ScriptRaccoon]

On the other hand, we don't need the functors themselves to carry data about the categories C and D in this context

Why? The same argument that applies to natural transformations (or morphisms in general) will also apply to functors, no?

[JoeLamond6]

My point is that in the context of the functor category [C,D], the functors are playing the role of the objects, not the arrows. But in the case of the arrows of [C,D] (i.e. natural transformations), we need to be able to recover the source and target of each arrow.

On the other hand, in the case of (say) the category of small categories, we do indeed need functors to carry the data of the categories, since the functors are playing the role of the arrows.

I know it feels ugly to have the formal definition of functor be different in different contexts, but all set-theoretic encodings are somewhat ugly ;)