When is a diagram cofibrant?

Emily Riehl – 15 February 2017

A natural question to ask for diagrams of a fixed shape valued in a model category is: when is the (weighted) colimit or limit functor invariant under weak equivalences? When this is the case, the strict (co)limit is referred to as a "homotopy (co)limit." In the case where the indexing category is a strict Reedy category, we present a notion of "Reedy cofibrant" diagrams and weights for which the strict weighted colimits are homotopy colimits. To illustrate this theory in practice, we then work out explicit characterizations of these Reedy cofibrant diagrams in the case of pushouts, coequalizers, sequential colimits, and geometric realizations.