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.
As this talk will explain, the "algebraic" aspects of the theory of Reedy categories very closely parallel the better known case of strict Reedy categories, described in the previous talks in this series, while the "homotopical" aspects are somewhat more complicated. In particular, we will introduce three notions of Reedy (algebraic) weak factorization system, and discuss their competing merits.