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A category $C$ can be seen as a topological space via the geometric realization of the nerve of the category. Then a functor of categories gives a map of spaces. Is there a nice categorical characterization of when functors give covering maps on the topological side (i.e. the map is a local homeomorphism)?

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4 Answers 4

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Coverings of groupoids are described for example in an ancient book http://www.maths.ed.ac.uk/~aar/papers/gz.pdf.

The condition on a groupoid functor C' -> C is that for any arrow f in C and an object x in C' which is taken to the source or the target of f by the given functor, there is a unique lifting of f in C' with the source or respectively the target x. For functors between categories the condition will be the same. (For groupoids in fact it is sufficient the condition to hold for sources only or for targets only).

I will add that, as mentioned, the geometric realization of a category goes through the nerve functor. The nerve functor is fully faithful, while simplicial sets which arise as nerves are characterized by the Segal condition. The covering functors are those which are taken to coverings of simplicial sets. The latter are charaterized in the Gabriel-Zisman's book. Then the above condition for a functor to be a covering is easily obtained using the Segal condition.

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You can find a very nice discussion of this question in the beginning of Quillen's paper on higher algebraic K-theory (Lecture Notes in Math #371). Quillen describes the category of covering spaces of BC as the category of "morphism-inverting" functors from C to sets. This is done using results of Gabriel-Zisman (see Dimitri Chikhladze's answer to this same question). The rough idea is that a functor from C to sets is describing the (isomorphic) discrete fibers over each zero-simplex in BC and then the Grothendieck construction (homotopy colimit in the sense of Thomason's thesis) applied to this functor actually builds a covering space.

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I think that the homotopical nature of the classifying space functor is such that it is unreasonable for it to return an actual covering space when applied to a functor. For instance, if you had a functor $f: C \to D$ where $Bf: BC \to BD$ was a covering, and then replaced $C$ by an equivalent category $C'$, you would get a homotopy equivalence $BC \simeq BC'$, but it would be unlikely that $BC' \to BD$ is itself a covering.

I think that the closest that you could actually hope for is that for a functor $f: C \to D$, the map $Bf: BC \to BD$ is a quasifibration with homotopy fibres which are homotopy equivalent to discrete spaces. I'll call this a "covering space up to homotopy."

Recall the hypotheses of Quillen's Theorem B: for any morphism $d' \to d$ the induced map of classifying spaces of comma categories $B (d' \setminus f) \to B(d \setminus f)$ is an equivalence; if this is the case, then the map $Bf: BD \to BC$ is a quasifibration with fibre $B(d \setminus f)$.

So in order to get a covering space up to homotopy, it would suffice to assume the hypotheses of Theorem B hold, and that all the comma categories $d \setminus f$ are all equivalent to sets (i.e., categories with only the identity morphisms). Then their classifying spaces are homotopically discrete.

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Thanks for your answer. The motivation for this question is the construction of the universal covering category as follows: Let $\pi_1(C)$ be the fundamental groupoid of $C$, i.e. the localization of $C$ on all arrows. Fix some object $X$ of $C$. Then let $UC$ be the slice catgory with respect to $X$ and the canonical $C\rightarrow\pi_1(C)$. Here rendiconti.dmi.units.it/volumi/25/25.pdf it is shown that it is simply conn. and I hoped I could easily see that it's also a covering of $C$. –  Werner Thumann Mar 7 at 15:19

Following on from Dimitri's answer, I first came across covering morphisms of groupoids in the paper

Higgins, P.J. "Presentations of groupoids, with applications to groups". Proc. Cambridge Philos. Soc. \60 (1964) 7--20.

and they are well presented in his book Categories and groupoids. In fact they appeared earlier in the paper

Smith, P.A. "The complex of a group relative to a set of generators. I". Ann. of Math. (2) 54 (1951) 371--402.

under the name "regular morphism".

One way of expounding the theory of covering spaces of a space $X$ is to give conditions on $X$ so that a particular covering morphism of groupoids $p: G \to \pi_1 X$ is determined by a covering map $q:Y \to X$ where $Y$ is a topologized $Ob(G)$.

March 9: I add that the category of covering morphisms of the groupoid $G$ is equivalent to the category of actions of $G$ on sets; so one can chose which of these you find best for a particular use or applications. Since 1967, I have liked modelling a covering map by a covering morphism, as was done for simplicial maps by Gabriel and Zisman.

You an find a full treatment of covering space theory using covering morphisms of groupoids in the book Topology and Groupoids, which extends somewhat the treatment in the 1968 edition.

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