The Stacks project

96.4 Sheaves

We first make an observation that is important and trivial (especially for those readers who do not worry about set theoretical issues).

Consider a big fppf site $\mathit{Sch}_{fppf}$ as in Topologies, Definition 34.7.6 and denote its underlying category $\mathit{Sch}_\alpha $. Besides being the underlying category of a fppf site, the category $\mathit{Sch}_\alpha $ can also can serve as the underlying category for a big Zariski site, a big étale site, a big smooth site, and a big syntomic site, see Topologies, Remark 34.11.1. We denote these sites $\mathit{Sch}_{Zar}$, $\mathit{Sch}_{\acute{e}tale}$, $\mathit{Sch}_{smooth}$, and $\mathit{Sch}_{syntomic}$. In this situation, since we have defined the big Zariski site $(\mathit{Sch}/S)_{Zar}$ of $S$, the big étale site $(\mathit{Sch}/S)_{\acute{e}tale}$ of $S$, the big smooth site $(\mathit{Sch}/S)_{smooth}$ of $S$, the big syntomic site $(\mathit{Sch}/S)_{syntomic}$ of $S$, and the big fppf site $(\mathit{Sch}/S)_{fppf}$ of $S$ as the localizations (see Sites, Section 7.25) $\mathit{Sch}_{Zar}/S$, $\mathit{Sch}_{\acute{e}tale}/S$, $\mathit{Sch}_{smooth}/S$, $\mathit{Sch}_{syntomic}/S$, and $\mathit{Sch}_{fppf}/S$ of these (absolute) big sites we see that all of these have the same underlying category, namely $\mathit{Sch}_\alpha /S$.

It follows that if we have a category $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ fibred in groupoids, then $\mathcal{X}$ inherits a Zariski, étale, smooth, syntomic, and fppf topology, see Stacks, Definition 8.10.2.

This definition makes sense by the discussion above. If $\mathcal{X}$ is an algebraic stack, the literature calls $\mathcal{X}_{fppf}$ (or a site equivalent to it) the big fppf site of $\mathcal{X}$ and similarly for the other ones. We may occasionally use this terminology to distinguish this construction from others.

Now that we have these topologies defined we can say what it means to have a sheaf on $\mathcal{X}$, i.e., define the corresponding topoi.

Of course we can also talk about sheaves of pointed sets, abelian groups, groups, monoids, rings, modules over a fixed ring, and lie algebras over a fixed field, etc. The category of abelian sheaves, i.e., sheaves of abelian groups, is denoted $\textit{Ab}(\mathcal{X}_{fppf})$ and similarly for the other topologies. If $\mathcal{X}$ is an algebraic stack, then $\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{fppf})$ is equivalent (modulo set theoretical problems) to what in the literature would be termed the category of sheaves on the big fppf site of $\mathcal{X}$. Similar for other topologies. We may occasionally use this terminology to distinguish this construction from others.

Since the topologies are listed in increasing order of strength we have the following strictly full inclusions

We sometimes write $\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{fppf}) = \mathop{\mathit{Sh}}\nolimits (\mathcal{X})$ and $\textit{Ab}(\mathcal{X}_{fppf}) = \textit{Ab}(\mathcal{X})$ in accordance with our terminology that a sheaf on $\mathcal{X}$ is an fppf sheaf on $\mathcal{X}$.

With this setup functoriality of these topoi is straightforward, and moreover, is compatible with the inclusion functors above.

In other words, pushforward and pullback of presheaves as defined in Section 96.3 also produces pushforward and pullback of $\tau $-sheaves. Having said all of the above we see that we can write $f^ p = f^{-1}$ and ${}_ pf = f_*$ without any possibility of confusion.

As discussed in Sites, Section 7.44 the same formula (on the underlying sheaf of sets) defines pushforward and pullback for sheaves (for one of our topologies) of pointed sets, abelian groups, groups, monoids, rings, modules over a fixed ring, and lie algebras over a fixed field, etc.