Post date: 08-oct-2015 23:24:42
In this talk we finally construct homotopy colimits via the bar construction. In the previous talk we observed that we have a natural (in F) weak equivalence from B(D,D,F) to F, we will show that by applying a pointwise cofibrant replacement if necessary to F, this constitutes a left deformation for the colimit functor. By the main theorem of the first talk, this proves that colim B(D,D,QF)=B(*,D,QF) is the homotopy colimit of F. The main ingredient to finish the proof is the theory of Reedy categories that gives us the homotopical properties of the bar construction, when combined with the yoga of tensor products the theorem will follow easily.