## Equivalences of complicial sets

### Alexander Campbell – 28 August 2019, 4 September 2019

In this talk, we will prove that a morphism of (weak) complicial sets is a marked homotopy equivalence iff (i) it is essentially surjective on (globular) n-cells for all n, and (ii) it reflects thinness of n-cells for all n; if the complicial sets are saturated, then property (i) suffices. It follows that a morphism of n-complicial sets is a marked homotopy equivalence iff it is essentially surjective on objects and a marked homotopy equivalence on (n-1)-complicial hom-sets.

The most difficult step in the proof is to show that every marked simplicial set can be expressed as an iterated homotopy colimit (in Verity's model structure for complicial sets) of the free-living n-cells. We will do so with the help of a combinatorial algorithm of Verity, which transmutes any n-simplex in a complicial set into one whose initial face is degenerate on a vertex.

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