0

The main refference: Prove that sheaf hom is a sheaf.

Without even knowing about the codomain of

$$\mathcal Hom(\mathscr F,\mathscr G):Open(X)^{op}\to (???)$$

Defining $\mathcal Hom(\mathscr F,\mathscr G)(U):=Nat(\mathscr F|_U,\mathscr G|_U)$ and proper restriction maps for $U\subset V$

$$\mathcal Hom(\mathscr F,\mathscr G)(V)\xrightarrow{\bar{res}_{V,U}} \mathcal Hom(\mathscr F,\mathscr G)(U)$$

One can show it is "a" sheaf on $X$. But I want to understand this $(???)$ the codomain of the functor $\mathcal Hom(\mathscr F,\mathscr G)$ so that I then can understand what happens if I change or take out $\mathscr F,\mathscr G$.

Objects of $(???)$ consist sets/classes of natural transformations between fixed restricted sheaves. But then I have seen Morphisms in the category of natural transformations?

So what is the proper way to understand $(???)$

Micheal Brain Hurts
  • 2,380
  • The sheaf hom is a sheaf. So Hom(F,G) has Set as its codomain (for example) – FShrike May 30 '23 at 16:56
  • what if we take sheaf of $\mathcal C$ not of sets? – Micheal Brain Hurts May 30 '23 at 16:57
  • Don't know what you mean. In the linked post hom(f,g) maps opens to sets of natural transformations – FShrike May 30 '23 at 16:59
  • 1
    Usually $\mathscr{Hom}(\mathscr{F}, \mathscr{G})$ is meant to be a sheaf of the same type as $\mathscr{F}$ and $\mathscr{G}$. So for example if $\mathscr{F},\mathscr{G}$ are sheaves of abelian group, your (???) would be $\mathbf{Ab}$. – Daniel Schepler May 30 '23 at 17:00
  • 3
    The needlessly general statement should (perhaps) be something like if $\mathcal{F},\mathcal{G}$ are sheaves valued in a category $\mathcal{C}$ enriched over a category $\mathcal{D}$, then $\mathcal{Hom}(\mathcal{F},\mathcal{G})$ is a sheaf valued in $\mathcal{D}$. Every category is enriched over $\mathbf{Set}$, so this is the "default" case as in FShrike's example. The category $\mathbf{Ab}$ is enriched over itself, so this also recovers Daniel Schepler's example and makes the sheaf hom in that case a so-called "internal Hom". – Thorgott May 30 '23 at 17:01
  • I dont know what enrichment is but, how you are so quickly determining the codomain, why $\mathcal Hom(\mathscr F,\mathscr G)(U):=Nat(\mathscr F|_U,\mathscr G|_U)$ becomes an abelian group if $\mathscr F,\mathscr G:Open(X)^{op}\to (Ab)$? I mean why sets of natural transformations as a set an abelian group? – Micheal Brain Hurts May 30 '23 at 17:18
  • 1
    One general situation where $\mathscr{Hom}$ is used is in the notion of a "monoidal closed category": this is a category $\mathbf{C}$ with a "tensor" functor $\mathbf{C}\times \mathbf{C} \to \mathbf{C}$ with some associativity conditions, and an "internal hom" functor $\mathbf{C}^{\operatorname{op}} \times \mathbf{C}\to\mathbf{C}$ - and a natural isomorphism ... – Daniel Schepler May 30 '23 at 18:22
  • 1
    $\operatorname{Hom}(X, \mathscr{Hom}(Y, Z)) \simeq \operatorname{Hom}(X\otimes Y, Z)$. – Daniel Schepler May 30 '23 at 18:23

1 Answers1

3

Thorgott's comment is correct, but I think looking at a few concrete examples may clarify your confusion.

In the case that $\mathscr{F,G}$ are sheaves of sets on a space $X$ (or any site, if you prefer/require such generality), then for each $U\subseteq X$ open, $\mathrm{Hom}(\mathscr{F}\vert_U,\mathscr{G}\vert_U)$ is a set, and so $\mathcal{H}\mathrm{om}(\mathscr{F},\mathscr{G})$ can be well-regarded as a sheaf of sets. The point is, the hom-sets in the category $\mathbf{Set}$ are again sets.

If $\mathscr{F,G}$ are sheaves of abelian groups, then the collection of sheaf homomorphisms $\mathrm{Hom}(\mathscr{F}\vert_U,\mathscr{G}\vert_U)$ will not just be a set, but will be equipped with the structure of an abelian group. So we can think of this "sheaf-hom" as a sheaf of abelian groups. This is because the category of sheaves of abelian groups is an abelian category, so its hom-sets are abelian groups.

This can be resolved in general by the theory of enriched categories, but I think any further generality will probably serve to confuse rather than illuminate. So I will just say this: whatever category your hom-sets belong to is the right target for the "sheaf-hom" just like in the above examples.

Charlie
  • 665