2-Final 2-functors

Abstract

A final functor between categories F:A→B is a functor that allows the restriction of diagrams on B to A without changing their colimits. More precisely, the functor F is final if, for any diagram D:B→E, there is a canonical isomorphismcolimBD≅colimAD∘F where either colimit exists whenever the other one does. There is a classical criterion for final functors [6, §IX.3]: a functor F:A→B is final if and only if, for any object b∈B, the slice category b/F is nonempty and connected. Such a criterion also exists for (∞,1)-categories, under the name of Quillen's Theorem A [5, §4.1]: an (∞,1)-functor F:A→B is final (with respect to any ∞-diagram) if and only if for any object b∈B, the slice ∞-category b/F is weakly contractible. One would expect a similar result for any dimension: an n-functor F:A→B is final (with respect to any n-diagram) if and only if, for any object b∈B, the slice n-category b/F is nonempty and has trivial homotopy groups πk for 0≤k≤n−1. Note that these are not consequences of the known criterion for 1-functors and (∞,1)-functors (see Remark 2.11 and Remark 3.3). A related result for 2-filtered 2-functors is proved in [1, §1.3].This paper presents a combinatorial proof in the case n=2 (Theorem 3.4). A general application is given in section 4, which can be specialized to 2-categories relevant to the study of finite groups.