In mathematics, especially in algebraic topology, the homotopy limit and colimitpg 52 are variants of the notions of limit and colimit extended to the homotopy category Ho ( Top ) {\displaystyle {\text{Ho}}({\textbf {Top}})} . The main idea is this: if we have a diagram F : I → Top {\displaystyle F:I\to {\textbf {Top}}} considered as an object in the homotopy category of diagrams F ∈ Ho ( Top I ) {\displaystyle F\in {\text{Ho}}({\textbf {Top}}^{I})} , (where the homotopy equivalence of diagrams is considered pointwise), then the homotopy limit and colimits then correspond to the cone and cocone Holim ← I ( F ) : ∗ → Top Hocolim → I ( F ) : ∗ → Top {\displaystyle {\begin{aligned}{\underset {\leftarrow I}{\text{Holim}}}(F)&:*\to {\textbf {Top}}\\{\underset {\rightarrow I}{\text{Hocolim}}}(F)&:*\to {\textbf {Top}}\end{aligned}}} which are objects in the homotopy category Ho ( Top ∗ ) {\displaystyle {\text{Ho}}({\textbf {Top}}^{*})} , where ∗ {\displaystyle *} is the category with one object and one morphism. Note this category is equivalent to the standard homotopy category Ho ( Top ) {\displaystyle {\text{Ho}}({\textbf {Top}})} since the latter homotopy functor category has functors which picks out an object in Top {\displaystyle {\text{Top}}} and a natural transformation corresponds to a continuous function of topological spaces. Note this construction can be generalized to model categories, which give techniques for constructing homotopy limits and colimits in terms of other homotopy categories, such as derived categories. Another perspective formalizing these kinds of constructions are derivatorspg 193 which are a new framework for homotopical algebra.
Introductory examples
Homotopy pushout The concept of homotopy colimitpg 4-8 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
D n ⊔ S n − 1 p t {\displaystyle D^{n}\sqcup _{S^{n-1}}pt}
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
p t ⊔ S n − 1 p t {\displaystyle pt\sqcup _{S^{n-1}}pt}
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 rectifies this defect. The homotopy pushout of two maps A ← B → C {\displaystyle A\leftarrow B\rightarrow C} of topological spaces is defined as
A ⊔ 1 B × [ 0 , 1 ] ⊔ 0 B ⊔ 1 B × [ 0 , 1 ] ⊔ 0 C {\displaystyle A\sqcup _{1}B\times [0,1]\sqcup _{0}B\sqcup _{1}B\times [0,1]\sqcup _{0}C} , 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 (whose maps are projections)
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