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.
Please help :)
