# Simplicial Covering Map

In Rezk's paper "A model for the homotopy theory of homotopy theory" numerous references to simplicial covering maps are made. It's first appearance being at the bottom of page 8. Unfortunately no definition is provided in the paper and I was wondering if there is a purely combinatorial definition for this concept or whether we have to pass to the geometric realization.

Maps of simplicial sets already match cells of the same dimension (roughly speaking), but it is the evenly covered concept that requires some work (I imagine).

Any help would be appreciated.

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I believe this is the fact you want: For a map of simplicial sets, in order for the map of realizations to be (topologically) isomorphic to $F\times U\to U$ with $F$ discrete for all small enough open sets $U$ of the codomain it is necessary and sufficient that the map is (simplicially) isomorphic to $F\times\sigma\to\sigma$ with $F$ discrete for every simplex $\sigma$ of the codomain. –  Tom Goodwillie Aug 24 '10 at 18:36
There is a theory of fiber bundles for simplical sets, which sometimes goes by the name of "twisted cartesian products": take a look at Chapter IV of May's Simplicial objects in algebraic topology, or Chapter V of Goerss-Jardine. Such bundles have "structure group" which is a simplical group $G$; the special case where $G$ is actually discrete gives you a theory of covering space. In which case you can reformulate as Tom describes. –  Charles Rezk Aug 24 '10 at 19:33
It looks like there should be a nice account of simplicial covering spaces in Gabriel & Zisman, Calculus of fractions and homotopy theory, though I don't have a copy hand to check exactly what they say. –  Charles Rezk Aug 24 '10 at 19:52
@Charles Rezk You are quite right: their treatment in Appendix 1.2 "Coverings of groupoids and simplicial coverings" inspired the treatment of covering spaces in the first 1968 edition of my topology book, now "Topology and groupoids" (2006). Though I omitted the simplicial set aspects! I particularly liked the equivalence between simplicial coverings and covering morphisms of the fundamental groupoid, and set out the corresponding topological theory. This is used in a recent paper of Brazas in HHA generalising covering space theory. –  Ronnie Brown Apr 27 '12 at 17:19