Determining the homotopy colimit of the following diagram.

Question

Here is the question :

1-Could anyone tell me how to calculate the pushout of the second row? I know that I will use Fubini theorem.

2- if someone describes for me the general procedure of answering those types of questions I would be very grateful.

Answer 1

One technique I likes to calculate a homotopy colimit is to break it down as a sequence of left homotopy Kan extensions. The result 3.11 in Groth’s paper “derivators, pointed derivators, monoidal derivators” is great for this, as it gives conditions under which this essentially reduces to calculating homotopy pushouts.

Speaking of which, with respect to your Question 1, to calculate a homotopy pushout you might use its local universal property. This says that maps from a homotopy pushout $P=A\sqcup_C B$ are given by maps from $A$ and $B$ together with a homotopy between their restriction to $C$. (This is equivalent to the model categorical prescription of taking a cofibrant replacement followed by a non-homotopy colimit.)

In the case at hand, the functor “two maps from the point and a homotopy between their restrictions to $S^0$” represented by the homotopy colimit can also be described at “a map from $S^0$ together with two nullhomotopies.” Whether in the based or unbased category, this is given by $S^1$.

Thus one calculates the Kan extension of your diagram along the inclusion into the category which extends each row to a square as having top and bottom rows all $*$, while the middle row is a suspension square for $S^0$. If this is all at all comprehensible without diagrams, perhaps you can see now how to finish up?