Post date: 17-sep-2015 2:27:46
Towards the aim of constructing homotopy colimits/limits we introduce the technology of enriched category theory, taking special care of simplicial enrichments. This is so because the bar construction of homotopy colimits uses the simplicial enrichment to state a "homotopical universal property". One could ask for a topological enrichment, but we will see that simplicial enrichments are more general. Simplicial categories are also interesting because they are models for ∞-categories, that is, there is a model structure on simplicial categories, and a Quillen equivalence with the Joyal model structure on simplicial sets.
In this talk we will introduce the basic yoga, including closed monoidal categories, V-adjunctions, change of base, the (weak) V-Yoneda lemma, tensors and cotensors. In the presence of a monoidal adjunction we will see that the change of base along the right adjoint transfer the tensors and cotensors, this is why we say that simplicial enrichments are more general, because we want the enrichment to be tensored and cotensored.