Morphisms of topoi preserve algebraic structure.

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

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

This method also works for sheaves of rings by thinking of a sheaf of rings (with unit) as a sextuple $(\mathcal{O}, + , 0, i, \cdot , 1)$ 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 $\lambda _ r \circ 0 = 0$, $\lambda _ r \circ + = + \circ (\lambda _ r, \lambda _ r)$, $\lambda _{r + r'} = + \circ \lambda _ r \times \lambda _{r'} \circ (\text{id}, \text{id})$, $\lambda _{rr'} = \lambda _ r \circ \lambda _{r'}$, $\lambda _1 = \text{id}$, $\lambda _0 = 0 \circ (\mathcal{F} \to *)$. $\square$

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