## Homotopy coherent structures, redux

### Emily Riehl – 26 July 2017

Naturally occurring diagrams in algebraic topology are commutative up to homotopy, but not on the nose. It was quickly realized that very little can be done with this information. Homotopy coherent category theory arose out of a desire to catalog the higher homotopical information required to restore constructibility (or more precisely, functoriality) in such "up to homotopy" settings. This talk will define homotopy coherent diagrams of topological spaces and prove that the free resolutions used to define homotopy coherent diagrams can also be understood as homotopy coherent realizations, i.e., via the left adjoint to the homotopy coherent nerve.

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