In this section we define presheaves on categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$, but most of the discussion works for categories over any base category. This section also serves to introduce the notation we will use later on.
Definition 96.3.1. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids.
A presheaf on $\mathcal{X}$ is a presheaf on the underlying category of $\mathcal{X}$.
A morphism of presheaves on $\mathcal{X}$ is a morphism of presheaves on the underlying category of $\mathcal{X}$.
We denote $\textit{PSh}(\mathcal{X})$ the category of presheaves on $\mathcal{X}$.
This defines presheaves of sets. Of course we can also talk about presheaves 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 presheaves, i.e., presheaves of abelian groups, is denoted $\textit{PAb}(\mathcal{X})$.
Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Recall that this means just that $f$ is a functor over $(\mathit{Sch}/S)_{fppf}$. The material in Sites, Section 7.19 provides us with a pair of adjoint functors1
The adjointness is
where $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\textit{PSh}(\mathcal{X}))$ and $\mathcal{G} \in \mathop{\mathrm{Ob}}\nolimits (\textit{PSh}(\mathcal{Y}))$. We call $f^ p\mathcal{G}$ the pullback of $\mathcal{G}$. It follows from the definitions that
for any $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X})$. The presheaf ${}_ pf\mathcal{F}$ is called the pushforward of $\mathcal{F}$. It is described by the formula
The rest of this section should probably be moved to the chapter on sites and in any case should be skipped on a first reading.
Lemma 96.3.2. Let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{Y} \to \mathcal{Z}$ be $1$-morphisms of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Then $(g \circ f)^ p = f^ p \circ g^ p$ and there is a canonical isomorphism ${}_ p(g \circ f) \to {}_ pg \circ {}_ pf$ compatible with adjointness of $(f^ p, {}_ pf)$, $(g^ p, {}_ pg)$, and $((g \circ f)^ p, {}_ p(g \circ f))$.
Proof. Let $\mathcal{H}$ be a presheaf on $\mathcal{Z}$. Then $(g \circ f)^ p\mathcal{H} = f^ p (g^ p\mathcal{H})$ is given by the equalities
We omit the verification that this is compatible with restriction maps.
Next, we define the transformation ${}_ p(g \circ f) \to {}_ pg \circ {}_ pf$. Let $\mathcal{F}$ be a presheaf on $\mathcal{X}$. If $z$ is an object of $\mathcal{Z}$ then we get a category $\mathcal{J}$ of quadruples $(x, f(x) \to y, y, g(y) \to z)$ and a category $\mathcal{I}$ of pairs $(x, g(f(x)) \to z)$. There is a canonical functor $\mathcal{J} \to \mathcal{I}$ sending the object $(x, \alpha : f(x) \to y, y, \beta : g(y) \to z)$ to $(x, \beta \circ f(\alpha ) : g(f(x)) \to z)$. This gives the arrow in
by Categories, Lemma 4.14.9. We omit the verification that this is compatible with restriction maps. An alternative to this direct construction is to define ${}_ p(g \circ f) \cong {}_ pg \circ {}_ pf$ as the unique map compatible with the adjointness properties. This also has the advantage that one does not need to prove the compatibility.
Compatibility with adjointness of $(f^ p, {}_ pf)$, $(g^ p, {}_ pg)$, and $((g \circ f)^ p, {}_ p(g \circ f))$ means that given presheaves $\mathcal{H}$ and $\mathcal{F}$ as above we have a commutative diagram
Proof omitted. $\square$
Lemma 96.3.3. Let $f, g : \mathcal{X} \to \mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $t : f \to g$ be a $2$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assigned to $t$ there are canonical isomorphisms of functors which compatible with adjointness of $(f^ p, {}_ pf)$ and $(g^ p, {}_ pg)$ and with vertical and horizontal composition of $2$-morphisms.
\[ t^ p : g^ p \longrightarrow f^ p \quad \text{and}\quad {}_ pt : {}_ pf \longrightarrow {}_ pg \]
Proof. Let $\mathcal{G}$ be a presheaf on $\mathcal{Y}$. Then $t^ p : g^ p\mathcal{G} \to f^ p\mathcal{G}$ is given by the family of maps
parametrized by $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X})$. This makes sense as $t_ x : f(x) \to g(x)$ and $\mathcal{G}$ is a contravariant functor. We omit the verification that this is compatible with restriction mappings.
To define the transformation ${}_ pt$ for $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{Y})$ define ${}_ y^ f\mathcal{I}$, resp. ${}_ y^ g\mathcal{I}$ to be the category of pairs $(x, \psi : f(x) \to y)$, resp. $(x, \psi : g(x) \to y)$, see Sites, Section 7.19. Note that $t$ defines a functor ${}_ yt : {}_ y^ g\mathcal{I} \to {}_ y^ f\mathcal{I}$ given by the rule
Note that for $\mathcal{F}$ a presheaf on $\mathcal{X}$ the composition of ${}_ yt$ with $\mathcal{F} : {}_ y^ f\mathcal{I}^{opp} \to \textit{Sets}$, $(x, f(x) \to y) \mapsto \mathcal{F}(x)$ is equal to $\mathcal{F} : {}_ y^ g\mathcal{I}^{opp} \to \textit{Sets}$. Hence by Categories, Lemma 4.14.9 we get for every $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{Y})$ a canonical map
We omit the verification that this is compatible with restriction mappings. An alternative to this direct construction is to define ${}_ pt$ as the unique map compatible with the adjointness properties of the pairs $(f^ p, {}_ pf)$ and $(g^ p, {}_ pg)$ (see below). This also has the advantage that one does not need to prove the compatibility.
Compatibility with adjointness of $(f^ p, {}_ pf)$ and $(g^ p, {}_ pg)$ means that given presheaves $\mathcal{G}$ and $\mathcal{F}$ as above we have a commutative diagram
Proof omitted. Hint: Work through the proof of Sites, Lemma 7.19.2 and observe the compatibility from the explicit description of the horizontal and vertical maps in the diagram.
We omit the verification that this is compatible with vertical and horizontal compositions. Hint: The proof of this for $t^ p$ is straightforward and one can conclude that this holds for the ${}_ pt$ maps using compatibility with adjointness. $\square$
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