# Mac Lane & Moerdijk's Exercise II.7.

This is Exercise II.7 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". According to the first few pages of this Approach0 question, it is new to MSE.

## The Details:

From p. 17 ibid. . . .

Definition 0: Given two functors

$$F:\mathbf{X}\to \mathbf{A}\quad G: \mathbf{A}\to \mathbf{X},$$

we say that $$G$$ is right adjoint to $$F$$, written $$F\dashv G$$, when for any $$X\in{\rm Ob}(\mathbf{X})$$ and any $$A\in{\rm Ob}(\mathbf{A})$$, there is a natural bijection between morphisms

$$\frac{X\stackrel{f}{\to}G(A)}{F(X)\stackrel{h}{\to}A},$$

in the sense that each $$f$$, as displayed, uniquely determines $$h$$, and conversely.

Adapted from p. 25, ibid. . . .

Definition 1: Let $$\mathbf{C}$$ be a category. Then $$\hat{\mathbf{C}}=\mathbf{Sets}^{\mathbf{C}^{{\rm op}}}$$ is the category of presheaves of $$\mathbf{C}$$.

From p. 79, ibid. . . .

Definition 2: For any space $$X$$, a continuous map $$p: Y\to X$$ is called a space over $$X$$ or a bundle over $$X$$.

From p. 83, ibid. . . .

Definition 3: [C]onsider any presheaf $$P: \mathcal{O}(X)^{{\rm op}} \to\mathbf{Sets}$$ on a space $$X$$, a point $$x$$, two open neighborhoods $$U$$ and $$V$$ of $$x$$, and two elements $$s \in PU$$, $$t \in PV$$. We say that $$s$$ and $$t$$ have the same germ at $$x$$ when there is some open set $$W \subseteq U\cap V$$ with $$x\in W$$ and $$s\rvert_E = t\rvert_W\in PW$$. This relation "has the same germ at $$x$$" is an equivalence relation, and the equivalence class of anyone such $$s$$ is called the germ of $$s$$ at $$x$$, in symbols $${\rm germ}_xs.$$

On p. 88 ibid. . . .

Definition 4: A bundle $$p: E \to X$$ is said to be étale (or étale over $$X$$) when $$p$$ is a local homeomorphism in the following sense: To each $$e\in E$$ there is an open set $$V$$, with $$e\in V\subset E$$, such that $$pV$$ is open in $$X$$ and $$p\rvert_V$$ is a homeomorphism $$V\to pV.$$

The functor $$\Gamma$$ is discussed here: . . . and what about $\Gamma$ in $\S II.5$ of Mac Lane and Moerdijk?

## The Question:

For any set $$T$$, the constant presheaf $$T$$ on a space $$X$$ has $$T(U) = T$$ for all open sets $$U$$ in $$X$$, with all restriction maps the identity. Show, using germs, that the associated étale space is the projection $$p: X \times T \to X$$ of the product, where $$T$$ has the discrete topology; conclude that the associated sheaf is the "constant" sheaf $$\Delta_T$$, for which $$\Delta_T(V)$$ is the set of all locally constant functions $$V \to T$$. Prove also that this defines a functor $$\Delta: \mathbf{Sets} \to {\rm Sh}(X)$$, which is left adjoint to the global sections functor $${\rm Sh}(X) \to \mathbf{Sets}, F \mapsto \Gamma F(X).$$

## Thoughts:

In order to give context, the following is a Q&A based on this meta question.

• What are you studying?

A postgraduate research degree in group theory. I'm studying topos theory for fun.

• What kind of approaches (to similar problems) are you familiar with?

This is novel to me, although I have worked with adjoint functors before. My approach here, then, is to write out the definitions as I have above with the intent of gaining an understanding of them and of how they come together to solve the problem. (So yeah . . . Pretty much how one would approach anything new. Sorry!)

• What kind of answer are you looking for?

An outline of an answer would be great. A full explanation would be much appreciated but I think it would spoil the fun.

• Is this question something you think should be able to answer? Why or why not?

I think, with a nudge in the right direction, that I could do this exercise myself; I just need to get started.

• As much as you work hard to put lots of background into your questions, this is still essentially just “how do I solve this problem?” Thus it’s not a very good question. Have you really not made any progress on your own? Note that all the things you end your question with are presented as alternatives to “what have you tried”, which should be the default. A “nudge in the right direction”, without more details about where you’re stuck, is naturally just going to be “use the definitions and the suggestion to prove the statement.” Commented Apr 5, 2020 at 3:07
• Honestly I agree with the comment above, and want to suggest that the most important question to answer when providing context is "Where are you getting stuck?" I look at this question, and I see you've written out all the definitions, but I can't see that you've tried to apply them to the question. Here's a rough outline of the steps you should think about. 1. Calculate the stalk at every point, 2. determine the topology on the etale space, 3. note that this is the same as the topology on $X\times T$, 4. Note that the associate sheaf consists of locally constant functions, 5. check adjunction
– jgon
Commented Apr 5, 2020 at 3:22
• This is basically outlined for you in the textbook question itself, so my question is, where do you get stuck?
– jgon
Commented Apr 5, 2020 at 3:22

The Étale space of a presheaf $$\mathcal F$$ on $$X$$ as a set is $$\bigsqcup_{x \in X} \mathcal F_x$$. Do you see why if $$\mathcal F = T(-)$$ then $$X \times T \approx \bigsqcup \mathcal F_x$$ as sets? Try and show that $$\mathcal F_x \approx T$$.
To prove adjointness it is clear that $$\Delta_T$$ is the sheafification of $$T(-)$$ by the above problem, therefore if we denote by $$\text{const}(T)$$ the constant presheaf with values in $$T$$ we have that $$\Delta = \text{sheafification} \circ \text{const}$$ and this simplifies the problem a lot since we know how to find right adjoints to composites of left adjoint functors in terms of their individual right adjoints.
Think about this for a second, if $$L$$ and $$L'$$ are two composable left adjoints with right adjoints $$R$$ and $$R'$$ then what is the right adjoint of $$L \circ L'$$?
The right adjoint to the sheafification functor is just the inclusion $$\text{Sheaf}(X) \rightarrow \text{Presheaf}(X)$$ but what is the right adjoint of $$\text{const}$$? Once you figure this out use the problem with the $$L$$ and $$L'$$ to find the right adjoint of $$\Delta$$.