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. 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.