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Homotopy colimit and limit
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<p> In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, especially in <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a>, the homotopy limit and colimitpg 52 are variants of the notions of <a href="/facts/Limit_(category_theory)/JoLugLs3">limit</a> and colimit extended to the homotopy category Ho ( Top ) {\displaystyle {\text{Ho}}({\textbf {Top}})} . The main idea is this: if we have a diagram</p><blockquote><p> F : I → Top {\displaystyle F:I\to {\textbf {Top}}} </p></blockquote><p>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 <a href="/facts/Limit_(category_theory)/JoLugLs3">cone</a> and cocone</p><blockquote><p> 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}}} </p></blockquote><p>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 <a href="/facts/Functor_category/r7sWdb2h">functor category</a> 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 <a href="/facts/Model_categories/KcxFUgY8">model categories</a>, which give techniques for constructing homotopy limits and colimits in terms of other homotopy categories, such as <a href="/facts/Derived_category/pBQHnFYo">derived categories</a>. Another perspective formalizing these kinds of constructions are <a href="/facts/Derivator/Rk0rcvKS">derivators</a>pg 193 which are a new framework for <a href="/facts/Homotopical_algebra/y0QtkpYr">homotopical algebra</a>. </p>