I'm currently at the very start of Sheaves in Geometry and Logic and the authors have showed that every sheaf is actually a sheaf of cross-sections.
I'll describe the definitions so I can state my question:
Sheaf of cross-sections: given a bundle $p: Y \to X$, $\Gamma_p: \mathcal{O}(X) \to Set$ is defined on $U$ as the set of all $s$ with $ps = i: U \to X$, the inclusion map.
The authors introduce germs around a point, that is, given a presheaf $P$, $s \in PU, t \in PW$ have same germ around $x$ if they agree on some open set around $x$. Thus the set $P_x$ of equivalence classes $\operatorname{germ}_x {s}$ for some $x$ is just the colimit of $P$ restricted to open sets containing $x$.
We then define $\Lambda_P$ to be the disjoint union of all $P_x$, and defining for each $s$ $\dot{s}$ to act taking $x$ to $\operatorname{germ}_x {s}$, we give $\Lambda_P$ a topology spanned by open images of the $\dot{s}$.
We then define the bundle $p: \Lambda_P \to X$ by taking $\operatorname{germ}_x {s}$ to $x$, and it turns out that for sheaves, the map
$$\eta_U: PU \to \Gamma_p \Lambda_PU \hspace{7mm} s \mapsto \dot{s}$$
is actually an isomorphism.
The authors also show that the right hand expression defines a functor which is a left adjoint to the inclusion $Sh(X) \hookrightarrow \widehat{\mathcal{O}(X)}$.
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Some time before picking the book I skimmed over an article named "an informal introduction to topos theory". The author gave a definition of $Sh(X)$ which I'd like to connect to this one.
Given a small category $\mathcal{C}$ and a functor to a cocomplete category $E: \mathcal{C} \to B$, we have an adjunction
$$\operatorname{colim}{(\int F \xrightarrow{\pi} C \xrightarrow{E} B)} \dashv \operatorname{Hom}{(E(\square), \underline{\space})} \; ,$$
where $\pi$ is the forgetful functor. I believe the left functor is usually denoted $\underline{\space} \otimes E$.
Now, setting $E$ to be $I: \mathcal{O}(X) \to \text{Top}/X$ taking $U$ to $U \hookrightarrow X$, the author states that $Sh(X)$ is precisely the subcategory whose restriction makes such adjunction an equivalence of categories, so that the units would be isomorphisms.
I'm trying to understand how these two definitions connect. For example, the units $\eta_P: P \to \operatorname{Hom}(I(\underline{\space}), P \otimes I)$ look similar to the previous $\eta_U$, in the sense that $(\eta_P)_U$ takes $PU$ to cross-sections w.r.t. the bundle $P \otimes I = Y_P \to X$ given by the colimit, that is, takes $PU$ to the set $\operatorname{Hom}_{\text{Top}/X}{(I(U), P \otimes I)}$.
However, I'm having a hard time "visualizing" what $P \otimes I = Y_P \to X$ should be. At first I thought I could identify $Y_P$ with $\Gamma \Lambda_P$, that is, that the constructions were identical, but I'm not sure how I'd do that.
My question is: how can I connect these two definitions?
I apologize if I made some mistake along the way, I'm new to the subject and trying to make sense of the definitions presented.
Best Answer
First of all, notice that the colimit of a diagram $F:D\rightarrow\mathbf{Top}/X$ can be calculated by taking the colimit $C$ of the functor $$D\rightarrow\mathbf{Top}/X\xrightarrow{U}\mathbf{Top}$$(where $U$ is the "forgetful functor"), then by universal property of colimits, $C$ comes with a unique map $C\rightarrow X$ making it into a colimit for the original diagram $F$.
Next, recall that given a functor $F:C^{op}\times C\rightarrow\mathcal S$ (with $C$ small and $\mathcal S$ cocomplete), the coend is calculated as the coequalizer of the two parallel maps $$\coprod_{d\rightarrow c}F(c,d)\rightrightarrows \coprod_cF(c,c)\tag{*}$$ that when precomposed with the canonical inclusion $F(x,y)\hookrightarrow \coprod_{d\rightarrow c}F(c,d)$ of the $(f:y\rightarrow x)$-factor into the disjoint union, we get the map $$F(x,y)\xrightarrow{F(f,1)}F(y,y)\hookrightarrow\coprod_cF(c,c)$$ and $$F(x,y)\xrightarrow{F(1,f)}F(x,x)\hookrightarrow\coprod_cF(c,c)$$
Now, consider the special case of $F=S\times I:(U,V)\mapsto S(U)\times I(V)$, with $S(U)$ having the discrete topology. As we said above, to calculate the coend of $F$ we can simply consider the induced functor $UF:C^{op}\times C\rightarrow\mathbf{Top}$ which is equal to $$(U,V)\mapsto S(U)\times V$$Then we can describe the coend diagram above in terms of elements: Elements of $\coprod_UF(U,U)$ are represented by pairs $(s,x)$, with $s\in S(U),x\in U$. Elements in $\coprod_{V\rightarrow U}F(U,V)$ are identified by pairs $(s,x)$, with $s\in S(U)$ and $x\in V$, for all inclusions $V\subseteq U$. Then the two maps in (*) acts as follows $$(s,x)\mapsto (s|_V,x),\qquad (s,x)\mapsto (s,x)$$(because the action of $V\subseteq U$ on $S(U)$ is restriction to $V$, while it's action on $I$ is just that of including $x\in V$ as an element of $U$).
The coend is defined as the quotient of $\coprod_UF(U,U)$ by the equivalence relation induced by the two maps $\iota_1,\iota_2$. In particular, notice that for all $x\in V\subseteq U$ and $s\in S(U)$, then $(s,x)\in S(U)\times U$ is identified with $(s|_V,x)\in S(V)\times V$, so if $s\in S(U),t\in S(V)$ are sections such that $s|_W=t|_W$, for some $x\in W\subseteq V\cap U$, then in the coend we have the following chain of equivalencies $$F(U)\times U\ni (s,x)\cong (s|_W,x)=(t|_W,x)\cong (t,x)\in F(V)\times V$$So, if $s,t$ have the same stalk at $x$, then $(s,x),(t,x)$ represent the same element in the coend.
This is the connection between the definition using stalks and coends: In both cases, you represent a point of $Y_S$, for a presheaf $S:C^{op}\rightarrow\mathcal Set$, as either a class $s_x\in S_x$(=stalk of $S$ at $x$), or an equivalence class of a pair $(s,x)$. Both $s_x$ and $(s,x)$ are meant to represent the value $s(x)\in Y_S$ of the "section" $s$ on $x$. What you do with stalks is first we identify when two sections $s,t$, defined on some point $x$, should have $s(x)=t(x)$, the answer is if $s,t$ are equal on the stalk $S_x$, so $S_x$ is the candidate for $\pi^{-1}(\{x\})$ (where $\pi:Y_S\rightarrow X$ is the projection), then we put all the fibers together with the appropriate topology and we get $Y_S$ the way Mac Lane does it. In the coend construction, the role of $s(x)$ is fulfilled by the equivalence class of $(s,x)\in F(U)\times U$. This time we first put together $S(U)$-many copies of $U$, for every open $U$, and then identify when $(s,x)$ and $(t,x)$ should represent the same element in $Y_S$. The end result are isomorphic spaces $Y_S\rightarrow X$ having $S$ as sheaf of sections.
This is my first time encountering the construction by coends (which is pretty neat in my opinion), so if there's any error with my interpretation of what is going on, do let me know.