Conditions on Group Actions

In this section we consider combinatorial conditions for a finite group G acting on a finite acyclic category C that ensure that taking the quotient of this group action commutes with the nerve functor. 

If Ac G AC is a group action on a category C, then A o Ac G — RTS is the associated group action on the nerve of C. It is clear that A C/G) is a sink for AoAc, see Subsection 4.4.1, and hence, as previously mentioned, the universal property of colimits guarantees the existence of a canonical map A A C)/G — AiCjG ). We wish to find conditions under which this map is an isomorphism. 

we prove in Proposition 14.11 that A is always surjective. Furthermore, G a) = [a] for a G 0(C), which means that, restricted to 0-skeletons, A is an isomorphism. If the two regular trisps were abstract simplicial complexes (only one face for any fixed vertex set), this would suffice to show isomorphism. Neither one is an abstract simplicial complex in general, but while the quotient of a complex A C)/G can have simplices with fairly arbitrary face sets in common, A C/G) has only one face for any fixed edge set. 

By the above description of A it suffices to fix a composable morphism chain ([mi]. [mt]) and to find a composable morphism chain (ni. nt), with [nj] = [mj]. The proof proceeds by induction on t. The case t = 1 is 

---