I am reading Moerdijk’s notes Introduction to the language of Stacks and Gerbes.

It says in page $4$ that

A sheaf of groups on $X$ is “the same” as an etale space $G\rightarrow X$, equipped with a continuously varying group structure on each of its fibers $G_x$ given by unit and multiplication maps $u:X\rightarrow G$ and $m:G\times G\rightarrow G$ (maps over $X$).

I do not understand the content that I have written in bold.

Given a sheaf $P$ on $X$, there is a notion of stalk of $P$ at $x\in X$ which is just the direct limit over $P(U)$ that contains $x$. In case when $P$ is a sheaf of groups, stalk is also a group.

With this in hand, consider $E=\bigsqcup_{x\in X}P_x$. This comes with a map $E\rightarrow X$ sending each element of $P_x$ to $x$. This gives a subjective map and we define topology on $ E$ to make this map continuous. It turns out that this map is actually a local hoemeomorphism, thus an etale space(map) $E\rightarrow X$.

See that fibers of this map are exactly $P_x$ and these fibers have group structure being direct limit of groups. I do not understand how they are seeing this structure to vary continuously.

I do not understand what this maps $u$ and $m$ are given for.

  • $\begingroup$ I do not know what these maps are. @EricWofsey $\endgroup$ – user537667 Mar 22 '18 at 6:41

First, $G$ is what you are calling $E$. The maps $u$ and $m$ are then literally just the group structures on all the $P_x$ put together. The map $u$ sends $x\in X$ to the identity element of the group $P_x$.

To understand $m$, we should first note that "$G\times G$" refers to the product in the category of (etale) spaces over $X$, so this is really the fiber product $G\times_X G=\{(a,b):a,b\in G,\pi(a)=\pi(b)\}$ where $\pi:G\to X$ is the projection. So an element of $G\times_X G$ is just a pair $(a,b)$ where $a$ and $b$ are both elements of $P_x$ for some $x$ (namely, $x=\pi(a)=\pi(b)$). The map $m$ then sends this pair $(a,b)$ to the product $ab\in P_x$ defined using the group structure on $P_x$.

To say the group structure "varies continuously" here just means that these maps $u$ and $m$ are continuous. For instance, to prove that $u$ is continuous, consider a basic open set $B$ in $G$. Such a basic open set is given by a section $s\in P(U)$ for some open $U\subseteq X$ and consists of all the images $s_x$ of $s$ in stalks $P_x$ for $x\in U$. The inverse image $u^{-1}(B)$ of this basic open set under $u$ is then just the set of $x\in U$ such that $s_x$ is the identity element of the group $P_x$. To prove this set is open, note that if $s_x$ is the identity element of $P_x$, then $s_x^2=s_x$. This implies there is some open set $V$ with $x\in V\subseteq U$ and $s|_V^2=s|_V$. But then this implies $s_y^2=s_y$ for all $y\in V$, so $s_y$ is the identity element of $P_y$ for all $y\in V$. That is, $V\subseteq u^{-1}(B)$, so $u^{-1}(B)$ contains a neighborhood of $x$.

The proof that $m$ is continuous is similar but a bit more complicated; again, the key step is that if an equation is true of the images of some sections in some stalk, it must be true when the sections are restricted to some open set.

Conversely, given such $u$ and $m$ which are continuous and define a group structure on each $P_x$, you can recover a group structure on the sheaf $P$ (actually, you just need $m$). Indeed, given sections $s,t\in P(U)$, we can think of these sections as continuous maps $s,t:U\to G$. We can then define the product $st:U\to G$ by $st(x)=m(s(x),t(x))$. With some work you can verify that this binary operation satisfies the groups operations and is compatible with the restrictions maps of $P$ and so makes $P$ a sheaf of groups.

A more elegant way to see all of this is to observe that the functor taking a sheaf to its etale space is an equivalence of categories between the category of sheaves (of sets) on $X$ and the category of etale spaces over $X$. A sheaf of groups on $X$ is just a group object in the category of sheaves: it is a sheaf $P$ together with maps $m:P\times P\to P$, $u:1\to P$, and $i:P\to P$ satisfying a categorical version of the group axioms for the product, identity, and inverse map of a group (here $1$ is the constant sheaf with value a 1-point set, the terminal object in the category of sheaves). Via our equivalence of categories, this is equivalent to the same structure in the category of etale spaces over $X$. This latter structure is almost what Moerdijk is describing, but for some reason he has omitted the inverse map $i$ and only mentioned $u$ and $m$. (This omission is essentially harmless in this case, since if maps $u:X\to G$ and $m:G\times G\to G$ are maps of etale spaces which define a group structure on each fiber of $G$ over $X$, then the inverse map $G\to G$ turns out to automatically be continuous.)

  • $\begingroup$ Thanks, this is what I was looking for. Can you suggest some supplementary material with Moerdijk notes for understanding about stacks and gerbes. I am reading simultaneously Angelo Vistoli’s notes on Grothendieck Topologies, fibered products and Descent theory. $\endgroup$ – user537667 Mar 22 '18 at 7:13
  • $\begingroup$ You Do you have any reference for notion of torsors in differential geometry set up? Angelo Vistoli’s noteshas some material on torsors but I do not fully understand. I am trying to understand arxiv.org/abs/math/0106083 $\endgroup$ – user537667 Mar 22 '18 at 16:19

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