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 $X$ be any topological space, $S$ - any category (e.g. of sets). Consider a new category $C$: its objects are only open subsets of $X$ and a set of morphisms from $U$ to $V$ is nonempty if and only if $U\subset V$. In this case we define the only one morphism $\phi_{UV}:U\rightarrow V$, that includes naturally $U$ into $V$. Thus we can define a presheaf of sets on $X$ as a contravariant functor $\Phi$ from $C$ to $S$ (that functor turns inclusion $U\subset V$ into restriction $res_{VU}$ - a morphism in $S$).
Perhaps, the following question is too popular, but i cannot get the answer myself.
Is it possible to define a sheaf in the same way? I mean some simple categorical terms.

share|cite|improve this question
Would sheafification be the sort of thing you're looking for? – Dan Rust May 7 '13 at 12:09

Yes, it is. If the target category is complete, the gluing axiom translates into $F(U) \rightarrow \prod_i F(U_i) \rightrightarrows \prod_{i,j} F(U_i \cap U_j)$ being a difference kernel diagram. This is actually a very common definition of sheafs, especially since it is no longer dependend on the starting category being the category of open sets of a topological space. On can (and does) define (pre)sheaves on any category in this way. This is especially useful when working with Grothendieck topologies etc.


share|cite|improve this answer
Little more detail can be found on Stacks Project -- Tag 0071. – Pece May 7 '13 at 12:18
"On can (and does) define (pre)sheaves on any category in this way." On sites, not arbitrary categories. Or are you refering to the canonical topology on a category? – Martin Brandenburg May 7 '13 at 13:24
What's a canonical topology on a category? – CSA Apr 21 '15 at 17:59
@CSA: "the", not "a" canonical topology. See… – Johannes Hahn Apr 21 '15 at 18:12

Your Answer


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