A counterexample in quasi-category theory

Abstract

We give an example of a morphism of simplicial sets which is a monomorphism, bijective on 0-simplices, and a weak categorical equivalence, but which is not inner anodyne.This answers an open question of Joyal. Furthermore, we use this morphism to refute a plausible description of the class of fibrations in Joyal's model structure for quasi-categories.

Generalizing quasi-categories via model structures on simplicial sets

“…For n = 2, the map on the left is actually an isomorphism since Λ S [2] is the union of two 1-simplices, one of which gets collapsed to a point to create Λ S [2] * i→i+1 . For n ≥ 3, we proceed by induction on |S|.…”

“…The complication is that in an arbitrary simplicial set, the set of I-edges do not necessarily satisfy the simplicial 2-out-of-3 property (unless I = ∆ [1]). As a minimal counter-example, one can simply take ∆ [2] itself and glue in a copy of I along two of its non-degenerate edges. The takeaway is that no single I can be used to identify which edges we want to view as invertible in an arbitrary simplicial set (except for the special case when I = ∆ [1]).…”

“…Let U I and V I be I-augmented unordered triangulations which the given edges factor through. Call the remaining edge e. Then we can define T I by gluing U I and V I to the appropriate faces of ∆ [2] as in Figure 2, making the remaining edge a T I -edge. Proposition 3.24.…”

“…Example 4.1. When X = ∆ [2], we glue in a copy of I along the three red edges of ∆ [1] × ∆ [2] depicted in Figure 3. Example 4.5.…”

“…For k > 2, there are no new edges added in the pushout, so the first claim is immediate. For k = 2, since the two edges of Λ j [2] are sent to almost-I-edges, the new edge is also an almost-I-edge in the pushout X ⊔ Λ j [2] ∆ [2] by applying the simplicial 2-out-of-3 property from Proposition 3.23. In each of these cases, the inclusion X ֒→ ∆[k] ⊔ Λ j [k] X can be upgraded to an almost-I-augmented horn pushout because every edge of the horn in X is an almost-I-edge.…”