**Title:** Algebraic perspectives on (generalized) Reedy categories

**Speaker:** Emily Riehl

**Speaker Info:** Harvard University

**Brief Description:**

**Special Note**:

**Abstract:**

Reedy categories are small categories equipped with a degree filtration on objects. The axioms allow for an inductive definition of diagrams and natural transformations, which is useful, in particular, for understanding the homotopical behavior of limits and colimits of diagrams with Reedy shape.The first part of this talk will summarize a recent expository paper with Dominic Verity. We give a canonical presentation of the hom bifunctor as a cell complex built from pushout-products of "boundary inclusions", which translates to a canonical presentation of any diagram or natural transformation as a (relative) cell complex and as a (relative) Postnikov tower whose cells are built from the latching or matching maps. This makes the proof of the Reedy model structure essentially trivial and leads to a geometric criterion characterizing the Reedy categories which give formulae for homotopy (co)limits.

The second part will report on work in progress to extend this theory to generalized Reedy categories: We propose that algebraic weak factorization systems are a natural tool to define the equivariant factorizations required to extend diagrams from one degree to the next.

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