Cell complex presentations for Reedy categories

Emily Riehl – 8 February 2017

A Reedy category (in the sense of Berger and Moerdijk, generalizing Kan's generalization of arguments due to Reedy for the category Delta) is a small category together with an assignment of a natural number degree to each of its objects together with designated subcategories of "degree-increasing" and "degree-decreasing" maps satisfying certain axioms. These axioms permit inductive specifications of the data of an object or morphism in the category of Reedy-indexed diagrams. In particular, these inductive arguments can be used to construct a Quillen model structure on the category of Reedy diagrams relative to any model structure on the target category, which can then be used to study homotopy limits or colimits of Reedy shape.

This expository talk will follow an expository paper joint with Verity that highlights the essential common feature of these inductive arguments. The Reedy axioms give rise to a canonical presentation of the hom bifunctor for any Reedy category, considered as a profunctor from that Reedy category to itself, as a cell complex whose cells are built from boundaries of representable functors. This structure is then transferred to generic Reedy diagrams by taking weighted limits or weighted colimits. Part II of this talk will consider the extension of these results from the strict Reedy case to the general one from the perspective of algebraic weak factorization systems.