The Stacks project

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

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

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 $\lambda _ r \circ 0 = 0$, $\lambda _ r \circ + = + \circ (\lambda _ r, \lambda _ r)$, $\lambda _{r + r'} = + \circ \lambda _ r \times \lambda _{r'} \circ (\text{id}, \text{id})$, $\lambda _{rr'} = \lambda _ r \circ \lambda _{r'}$, $\lambda _1 = \text{id}$, $\lambda _0 = 0 \circ (\mathcal{F} \to *)$. $\square$


Comments (1)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 00YV. Beware of the difference between the letter 'O' and the digit '0'.