Abstract. Consider a push-out diagram of spaces C ← A → B, construct the homotopy push-out, and then the homotopy pull-back of the diagram one gets by forgetting the initial object A. We compare the difference between A and this homotopy pull-back. This difference is measured in terms of the homotopy fibers of the original maps. Restricting our attention to the connectivity of these maps, we recover the classical Blakers-Massey Theorem.
We answer the question to what extent homotopy (co)limits in categories with weak equivalences allow for a Fubini-type interchange law. The main obstacle is that we do not assume our categories with weak equivalences to come equipped with a calculus for homotopy (co)limits, such as a derivator.Proposition 8.1. Let C be a category and J an index category. For J ∈ J with inclusion In J : {J} ֒→ J, the evaluation functor ev J : C J → C has a right adjoint(with the obvious arrow map), granted all these powers in C exist. Moreover, if End J (J) = {id J }, then J * is fully faithful or equivalently, a counit ε of the adjunction is invertible. Dually, ev J has a left adjointgranted all these copowers exist and if End J (J) = {id J }, then J ! is fully faithful.Proof. Note that ev J = J * is given by precomposition with In J : {J} ֒→ J and thus has a right adjoint given by taking the pointwise right Kan extension along J (granted that all these exist)Example 8.2. For J any direct or inverse category (in the sense of the theory of Reedy categories), then J ! and J * (assuming they exist) are fully faithful for all J ∈ J. In particular, this is true for the indexing categories of the most common homotopy colimits, such as homotopy pushouts, coproducts, telescopes and coequalisers.
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