Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $\mathcal F$ be a sheaf of sets on a site. Fix an object $X$ of the underlying category of the site, which is assumed to contain a final object and have products. Define a presheaf $\mathcal G$ by setting $\mathcal G(U) := \mathcal F(U \times X)$. Is $\mathcal G$ a sheaf?

share|cite|improve this question

If $(U_i)$ cover $U$, then $\mathcal{G}(U)$ is supposed to be a certain limit of a diagram of the $\mathcal{G}(U_i)$ and $\mathcal{G}(U_i \times_U U_j)$. The $U_i \times X$ also cover $U \times X$ (this is the base change axiom for the site), and $(U_i \times_U U_j) \times X = (U_i \times X) \times_{U \times X} (U_j \times X)$, so the analogous diagram for $\mathcal{F}(U \times X)$ is indeed a limit diagram, by the sheaf condition on $\mathcal{F}$. Thus the answer is yes.

share|cite|improve this answer

Your operation can be defined intrinsically without reference to sites: if $\mathcal{E}$ is your topos and $X$ is an object in $\mathcal{E}$, the slice category $\mathcal{E}_{/X}$ is again a topos, and there is a geometric morphism $p^* \dashv p_* : \mathcal{E}_{/X} \to \mathcal{E}$ where the inverse image functor $p^*$ is the functor $- \times X$, and $p^*$ itself has a left adjoint $p_! : \mathcal{E}_{/X} \to \mathcal{E}$, which is just the evident projection functor.

I claim the operation you describe is the endofunctor $p_* p^* : \mathcal{E} \to \mathcal{E}$. Indeed, by the Yoneda lemma, if $U$ and $X$ are representable sheaves in $\mathcal{E}$ and the site has products, then, $$p_* p^* \mathscr{F} (U) = \mathcal{E}(U, p_* p^* \mathscr{F}) \cong \mathcal{E}_{/X}(p^* U, p^* \mathscr{F}) \cong \mathcal{E} (p_! p^* U, \mathscr{F})$$ but $p_! p^* U = U \times X$, so $$p_* p^* \mathscr{F} (U) \cong \mathscr{F} (U \times X)$$ as claimed.

share|cite|improve this answer
This may be a dumb question, but isn't $X$ supposed to be an object of the site corresponding to $\mathcal{E}$? – Martin Brandenburg Mar 20 '13 at 14:07
Yes, but if the site has a subcanonical topology you can just embed the site as a full subcategory of $\mathcal{E}$ via Yoneda. If not then you can take the sheafification. – Zhen Lin Mar 20 '13 at 16:18

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.