Higher zeroes in pointed infinity-categories

Abstract

We construct an endofunctor on (small) simplicial sets which freely adjoins a zero object to a (small) infinity-category, as the co-cone construction freely adjoins a terminal object. We then prove that this endofunctor induces a left adjoint to the inclusion of the infinity-category of (small) pointed infinity-categories and morphisms that preserve zero objects into the infinity-category of (small) infinity-categories. This adjunction, in turn, leads to the definition of a coherent choice of nullhomotopies on a pointed infinity-category. A coherent choice of nullhomotopies is a non-canonical, coherent choice of "higher zeroes" on a pointed infinity-category, mirroring the obvious definition of higher zeroes on the differential graded nerve of a differential graded category of chain complexes over an additive category. We then show that coherent choices of nullhomotopies give an alternate characterization of fiber sequences in pointed infinity-categories, generalizing the 1-categorical definition of kernels in terms of morphisms that compose to zero.