The Stacks project

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7.44 Sheaves of algebraic structures

In Sheaves, Section 6.15 we introduced a type of algebraic structure to be a pair $(\mathcal{A}, s)$, where $\mathcal{A}$ is a category, and $s : \mathcal{A} \to \textit{Sets}$ is a functor such that

  1. $s$ is faithful,

  2. $\mathcal{A}$ has limits and $s$ commutes with limits,

  3. $\mathcal{A}$ has filtered colimits and $s$ commutes with them, and

  4. $s$ reflects isomorphisms.

For such a type of algebraic structure we saw that a presheaf $\mathcal{F}$ with values in $\mathcal{A}$ on a space $X$ is a sheaf if and only if the associated presheaf of sets is a sheaf. Moreover, we worked out the notion of stalk, and given a continuous map $f : X \to Y$ we defined adjoint functors pushforward and pullback on sheaves of algebraic structures which agrees with pushforward and pullback on the underlying sheaves of sets. In addition extending a sheaf of algebraic structures from a basis to all opens of a space, works as expected.

Part of this material still works in the setting of sites and sheaves. Let $(\mathcal{A}, s)$ be a type of algebraic structure. Let $\mathcal{C}$ be a site. Let us denote $\textit{PSh}(\mathcal{C}, \mathcal{A})$, resp. $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}, \mathcal{A})$ the category of presheaves, resp. sheaves with values in $\mathcal{A}$ on $\mathcal{C}$.

  • ($\alpha $) A presheaf with values in $\mathcal{A}$ is a sheaf if and only if its underlying presheaf of sets is a sheaf. See the proof of Sheaves, Lemma 6.9.2.

  • ($\beta $) Given a presheaf $\mathcal{F}$ with values in $\mathcal{A}$ the presheaf ${\mathcal{F}}^\# = (\mathcal{F}^+)^+$ is a sheaf. This is true since the colimits in the sheafification process are filtered, and even colimits over directed sets (see Section 7.10, especially the proof of Lemma 7.10.14) and since $s$ commutes with filtered colimits.

  • ($\gamma $) We get the following commutative diagram

    \[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}, \mathcal{A}) \ar@<1ex>[r] \ar[d]_ s & \textit{PSh}(\mathcal{C}, \mathcal{A}) \ar@<1ex>[l]^{{}^\# } \ar[d]^ s\\ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \ar@<1ex>[r] & \textit{PSh}(\mathcal{C}) \ar@<1ex>[l] } \]
  • ($\delta $) We have $\mathcal{F} = \mathcal{F}^\# $ if and only if $\mathcal{F}$ is a sheaf of algebraic structures.

  • ($\epsilon $) The functor ${}^\# $ is adjoint to the inclusion functor:

    \[ \mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{C}, \mathcal{A})}(\mathcal{G}, \mathcal{F}) = \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C}, \mathcal{A})}(\mathcal{G}^\# , \mathcal{F}) \]

    The proof is the same as the proof of Proposition 7.10.12.

  • ($\zeta $) The functor $\mathcal{F} \mapsto \mathcal{F}^\# $ is left exact. The proof is the same as the proof of Lemma 7.10.14.

Definition 7.44.1. Let $f : \mathcal{D} \to \mathcal{C}$ be a morphism of sites given by a functor $u : \mathcal{C} \to \mathcal{D}$. We define the pushforward functor for presheaves of algebraic structures by the rule $u^ p\mathcal{F}(U) = \mathcal{F}(uU)$, and for sheaves of algebraic structures by the same rule, namely $f_*\mathcal{F}(U) = \mathcal{F}(uU)$.

The problem comes with trying the define the pullback. The reason is that the colimits defining the functor $u_ p$ in Section 7.5 may not be filtered. Thus the axioms above are not enough in general to define the pullback of a (pre)sheaf of algebraic structures. Nonetheless, in almost all cases the following lemma is sufficient to define pushforward, and pullback of (pre)sheaves of algebraic structures.

Lemma 7.44.2. Suppose the functor $u : \mathcal{C} \to \mathcal{D}$ satisfies the hypotheses of Proposition 7.14.7, and hence gives rise to a morphism of sites $f : \mathcal{D} \to \mathcal{C}$. In this case the pullback functor $f^{-1}$ (resp. $u_ p$) and the pushforward functor $f_*$ (resp. $u^ p$) extend to an adjoint pair of functors on the categories of sheaves (resp. presheaves) of algebraic structures. Moreover, these functors commute with taking the underlying sheaf (resp. presheaf) of sets.

Proof. We have defined $f_* = u^ p$ above. In the course of the proof of Proposition 7.14.7 we saw that all the colimits used to define $u_ p$ are filtered under the assumptions of the proposition. Hence we conclude from the definition of a type of algebraic structure that we may define $u_ p$ by exactly the same colimits as a functor on presheaves of algebraic structures. Adjointness of $u_ p$ and $u^ p$ is proved in exactly the same way as the proof of Lemma 7.5.4. The discussion of sheafification of presheaves of algebraic structures above then implies that we may define $f^{-1}(\mathcal{F}) = (u_ p\mathcal{F})^\# $. $\square$

We briefly discuss a method for dealing with pullback and pushforward for a general morphism of sites, and more generally for any morphism of topoi.

Let $\mathcal{C}$ be a site. In the case $\mathcal{A} = \textit{Ab}$, we may think of an abelian (pre)sheaf on $\mathcal{C}$ as a quadruple $(\mathcal{F}, +, 0, i)$. Here the data are

  1. $\mathcal{F}$ is a sheaf of sets,

  2. $+ : \mathcal{F} \times \mathcal{F} \to \mathcal{F}$ is a morphism of sheaves of sets,

  3. $0 : * \to \mathcal{F}$ is a morphism from the singleton sheaf (see Example 7.10.2) to $\mathcal{F}$, and

  4. $i : \mathcal{F} \to \mathcal{F}$ is a morphism of sheaves of sets.

These data have to satisfy the following axioms

  1. $+$ is associative and commutative,

  2. $0$ is a unit for $+$, and

  3. $+ \circ (1, i) = 0 \circ (\mathcal{F} \to *)$.

Compare Sheaves, Lemma 6.4.3. Let $f : \mathcal{D} \to \mathcal{C}$ be a morphism of sites. Note that since $f^{-1}$ is exact we have $f^{-1}* = *$ and $f^{-1}(\mathcal{F} \times \mathcal{F}) = f^{-1}\mathcal{F} \times f^{-1}\mathcal{F}$. Thus we can define $f^{-1}\mathcal{F}$ simply as the quadruple $(f^{-1}\mathcal{F}, f^{-1}+, f^{-1}0, f^{-1}i)$. The axioms are going to be preserved because $f^{-1}$ is a functor which commutes with finite limits. Finally it is not hard to check that $f_*$ and $f^{-1}$ are adjoint as usual.

In [SGA4] this method is used. They introduce something called an “espèce the structure algébrique $\ll $définie par limites projectives finie$\gg $”. For such an espèce you can use the method described above to define a pair of adjoint functors $f^{-1}$ and $f_*$ as above. This clearly works for most algebraic structures that one encounters in practice. Instead of formalizing this construction we simply list those algebraic structures for which this method works (to be verified case by case). In fact, this method works for any morphism of topoi.

slogan

Proposition 7.44.3. Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $f = (f^{-1}, f_*)$ be a morphism of topoi from $\mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$. The method introduced above gives rise to an adjoint pair of functors $(f^{-1}, f_*)$ on sheaves of algebraic structures compatible with taking the underlying sheaves of sets for the following types of algebraic structures:

  1. pointed sets,

  2. abelian groups,

  3. groups,

  4. monoids,

  5. rings,

  6. modules over a fixed ring, and

  7. lie algebras over a fixed field.

Moreover, in each of these cases the results above labeled ($\alpha $), ($\beta $), ($\gamma $), ($\delta $), ($\epsilon $), and ($\zeta $) hold.

Proof. The final statement of the proposition holds simply since each of the listed categories, endowed with the obvious forgetful functor, is indeed a type of algebraic structure in the sense explained at the beginning of this section. See Sheaves, Lemma 6.15.2.

Proof of (2). We think of a sheaf of abelian groups as a quadruple $(\mathcal{F}, +, 0, i)$ as explained in the discussion preceding the proposition. If $(\mathcal{F}, +, 0, i)$ lives on $\mathcal{C}$, then its pullback is defined as $(f^{-1}\mathcal{F}, f^{-1}+, f^{-1}0, f^{-1}i)$. If $(\mathcal{G}, +, 0, i)$ lives on $\mathcal{D}$, then its pushforward is defined as $(f_*\mathcal{G}, f_*+, f_*0, f_*i)$. This works because $f_*(\mathcal{G} \times \mathcal{G}) = f_*\mathcal{G} \times f_*\mathcal{G}$. Adjointness follows from adjointness of the set based functors, since

\[ \mathop{Mor}\nolimits _{\textit{Ab}(\mathcal{C})} ((\mathcal{F}_1, +, 0, i), (\mathcal{F}_2, +, 0, i)) = \left\{ \begin{matrix} \varphi \in \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})} (\mathcal{F}_1, \mathcal{F}_2) \\ \varphi \text{ is compatible with }+, 0, i \end{matrix} \right\} \]

Details left to the reader.

This method also works for sheaves of rings by thinking of a sheaf of rings (with unit) as a sixtuple $(\mathcal{O}, + , 0, i, \cdot , 1)$ satisfying a list of axioms that you can find in any elementary algebra book.

A sheaf of pointed sets is a pair $(\mathcal{F}, p)$, where $\mathcal{F}$ is a sheaf of sets, and $p : * \to \mathcal{F}$ is a map of sheaves of sets.

A sheaf of groups is given by a quadruple $(\mathcal{F}, \cdot , 1, i)$ with suitable axioms.

A sheaf of monoids is given by a pair $(\mathcal{F}, \cdot )$ with suitable axiom.

Let $R$ be a ring. An sheaf of $R$-modules is given by a quintuple $(\mathcal{F}, +, 0, i, \{ \lambda _ r\} _{r \in R})$, where the quadruple $(\mathcal{F}, +, 0, i)$ is a sheaf of abelian groups as above, and $\lambda _ r : \mathcal{F} \to \mathcal{F}$ is a family of morphisms of sheaves of sets such that $\lambda _ r \circ 0 = 0$, $\lambda _ r \circ + = + \circ (\lambda _ r, \lambda _ r)$, $\lambda _{r + r'} = + \circ \lambda _ r \times \lambda _{r'} \circ (\text{id}, \text{id})$, $\lambda _{rr'} = \lambda _ r \circ \lambda _{r'}$, $\lambda _1 = \text{id}$, $\lambda _0 = 0 \circ (\mathcal{F} \to *)$. $\square$

We will discuss the category of sheaves of modules over a sheaf of rings in Modules on Sites, Section 18.10.

Remark 7.44.4. Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $u : \mathcal{D} \to \mathcal{C}$ be a continuous functor which gives rise to a morphism of sites $\mathcal{C} \to \mathcal{D}$. Note that even in the case of abelian groups we have not defined a pullback functor for presheaves of abelian groups. Since all colimits are representable in the category of abelian groups, we certainly may define a functor $u_ p^{ab}$ on abelian presheaves by the same colimits as we have used to define $u_ p$ on presheaves of sets. It will also be the case that $u_ p^{ab}$ is adjoint to $u^ p$ on the categories of abelian presheaves. However, it will not always be the case that $u_ p^{ab}$ agrees with $u_ p$ on the underlying presheaves of sets.


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