Functors (between oo-categories) that aren't strictly unital

TL;DR

This paper investigates when a structure-preserving assignment between quasi-categories can be refined into a genuine functor, showing that respecting identities up to homotopy suffices for such a refinement.

Contribution

It establishes conditions under which a diagram-respecting assignment between infinity-categories can be promoted to an actual functor, respecting identities up to homotopy.

Findings

Assignments respecting face maps can be extended to functors if identities are respected up to homotopy.

Such functors can be chosen to vary naturally with the original assignments.

The results clarify when non-strict functoriality can be rectified in infinity-category theory.

Abstract

Let C and D be quasi-categories (a.k.a. infinity-categories). Suppose also that one has an assignment sending commutative diagrams of C to commutative diagrams of D which respects face maps, but not necessarily degeneracy maps. (This is akin to having an assignment which respects all compositions, but may not send identity morphisms to identity morphisms.) When does this assignment give rise to an actual functor? We show that if the original assignment can be shown to respect identity morphisms up to homotopy, then there exists an honest functor of infinity-categories which respects the original assignments up to homotopy. Moreover, we prove that such honest functors can be chosen naturally with respect to the original assignments.