Homotopy colimit and limit

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.

Source: Wikipedia — Homotopy colimit and limit (CC BY-SA 4.0)