As the title says, I want to show that for any topological space , the category of set-valued presheaves PSh(X) on has all finite limits and co-limits.
First, PSh(X) has both initial and terminal objects. With a bit of thought, these are (obviously) the constant functors 0 and 1 (resp.) where, for all open subsets of we have and and the morphisms for 0 and 1 are just the identities.
Second, we need to show that PSh(X) has a pullback for every diagram and a pushout for every diagram .
Indeed, let be a diagram of presheaves. I claim that the presheaf defined via for all open. Basically, we just define the pullback “element-wise” on the source category. The morphisms are a bit tricky though. Indeed, given a map of open sets, how do we define the “restriction” map ? One should never diagram chase in public, but let me assure you that it really ends up just being the map guaranteed by the universal property of the pullback.
(picture for clarity). It’s a bit clumsy, but one gets a pushout in the same manner, defining it element-wise as a the quotient , where iff there exists a such that and . This sometimes denoted as .
Since PSh(X) has a terminal object and pullbacks, it has a finite limits. On the other hand, it has an initial object and pushouts, so it has all finite co-limits. Easy.
It turns out that the category of sheaves on , Sh(X) is finitely bi-complete as well, but showing it for PSh(X) is really easy so I decided to just do that one. Next time, I want to look at the following case: If there is pair of adjoint functors and with , do we have a pair of adjoint functors between sheaves with values in C and sheaves with values in D on some topological space ? I’ve only looked at the case where the adjunction is the forgetful functor and free abelian group functor and the case of presheaves, where this statement does in fact hold. Until next time. 🙂