Homotopy colimit

In mathematics, especially in algebraic topology, the homotopy limit and colimit are variants of the notions of limit and colimit. They are denoted by holim and hocolim, respectively.

The concept of homotopy colimit is a generalization of homotopy pushouts, such as the mapping cylinder used to define a cofibration. This notion is motivated by the following observation: the (ordinary) pushout

is the space obtained by contracting the n-1-sphere (which is the boundary of the n-dimensional disk) to a single point. This space is homeomorphic to the n-sphere Sⁿ. On the other hand, the pushout

is a point. Therefore, even though the (contractible) disk Dⁿ was replaced by a point, (which is homotopy equivalent to the disk), the two pushouts are not homotopy (or weakly) equivalent.

Therefore, the pushout is not well-aligned with a principle of homotopy theory, which considers weakly equivalent spaces as carrying the same information: if one (or more) of the spaces used to form the pushout is replaced by a weakly equivalent space, the pushout is not guaranteed to stay weakly equivalent. The homotopy pushout does not share this defect.

The homotopy pushout of two maps ${\displaystyle A\leftarrow B\rightarrow C}$ of topological spaces is defined as

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