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 pushout is the space obtained by contracting the n-1-sphere 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 disk Dn was replaced by a point,, the two pushouts are not homotopy 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 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 rectifies this defect. The homotopy pushout of two maps of topological spaces is defined as i.e., instead of glueing B in both A and C, two copies of a cylinder on B are glued together and their ends are glued to A and C. For example, the homotopy colimit of the diagram is the join. It can be shown that the homotopy pushout does not share the defect of the ordinary pushout: replacingA, B and / or C by a homotopic space, the homotopy pushout will also be homotopic. In this sense, the homotopy pushouts treats homotopic spaces as well as the pushout does with homeomorphic spaces.
Mapping telescope
The homotopy colimit of a sequence of spaces is the mapping telescope.
Treating examples such as the mapping telescope and the homotopy pushout on an equal footing can be achieved by considering an -diagram of spaces, where is some "indexing" category. This is a functor i.e., to each object in, one assigns a space and maps between them, according to the maps in. The category of such diagrams is denoted. There is a natural functor called the diagonal, which sends any space to the diagram which consists of everywhere. In category theory, the right adjoint to this functor is the limit. The homotopy limit is defined by altering this situation: it is the right adjoint to which sends a space to the -diagram which at some object gives Here is the slice category, is the nerve of this category and |-| is the topological realization of this simplicial set.
Homotopy colimit
Similarly, one can define a colimit as the left adjoint to the diagonal functor given above. To define a homotopy colimit, we must modify in a different way. A homotopy colimit can be defined as the left adjoint to a functor where where is the opposite category of. Although this is not the same as the functor above, it does share the property that if the geometric realization of the nerve category is replaced with a point space, we recover the original functor.
Relation to the (ordinary) colimit and limit
There is always a map Typically, this map is not a weak equivalence. For example, the homotopy pushout encountered above always maps to the ordinary pushout. This map is not typically a weak equivalence, for example the join is not weakly equivalent to the pushout of, which is a point.
