# Putting sheaves to work for algebraic topology?

This is cross-posted from math.se after receiving points and no answers. I apologise if this question is too basic for MathOverflow.

I'm refreshing my memory of covering space theory, and this time around, I know some sheaf theory. It feels like arguments are used to prove results about covering spaces, such as uniqueness of lifts, having something "sheafy" about them.

For example, to prove uniqueness of lifts, we argue by trying to extend "equality at a point" to "equality over a neighbourhood" to "equality over the entire domain". It seems like the language of sheaves may make this clearer?

Similarly, when it comes to covering spaces, there is something "etale-like" about them. Is there a reference that expands on this perspective?

• The most general version of this I know is this paper, but there's a huge gulf of successive abstractions from covering space theory to that :). – Denis Nardin Mar 7 at 22:33
• In the PhD thesis of Ingo Blechschmidt the internal language of sheaves over a space is developed to treat the questions of when is it possible to extend 'truth at points' to 'truths over a neighbourhood' (which is global truth for sheaves, since truth glues). – mattecapu Mar 8 at 10:48

## 1 Answer

For sufficiently nice topological spaces $$X$$ (e.g., locally connected for the last two categories to be equivalent, and semilocally simply connected and locally path-connected for all three to be equivalent), the following three categories are equivalent:

• Functors from the fundamental groupoid of $$X$$ to the category of sets;

• Covering spaces over $$X$$;

• Locally constant sheaves of sets on $$X$$.

This is an extremely primitive baby version of the Riemann–Hilbert correspondence.

References specifically for this elementary case are sparse, but there is an extensive discussion on locally constant sheaves at the nCafé.

• Aren't the last two categories always equivalent (at least if you take the right definition of covering spaces as fiber bundles with discrete fibers)? – Denis Nardin Mar 8 at 7:32
• A complete discussion of the equivalence between the first two categories (in german) is in the topology textbook by Laures and Szymik. – Konrad Waldorf Mar 8 at 9:23
• This is a minor comment, but to my mind all three of those categories live on the same side of the Riemann-Hilbert correspondence. The other side would consist of some form of flat connections (which only really makes sense in a differential/algebro-geometric context with linear coefficients). I guess I would call the equivalence of the first two categories the Galois correspondence or something. – Sam Gunningham Mar 8 at 15:04
• @DenisNardin: What other definition of a covering space do you have in mind? From my point of view, the problem is that there are two nonequivalent definitions of a locally constant sheaf, and the one commonly given in the literature (a sheaf that is locally isomorphic to the constant sheaf) simply does not give a category equivalent to covering spaces, as explained in the linked nCafé post. – Dmitri Pavlov Mar 8 at 15:04
• @SamGunningham: Covering spaces resemble flat connections: both admit a notion of a parallel transport that is invariant under homotopies of paths. Both have a total space that projects to the base space. – Dmitri Pavlov Mar 8 at 15:05