Homotopy colimit and limit

rdf:langString Homotopy colimit and limit

rdf:langString 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 . The main idea is this: if we have a diagram considered as an object in the , (where the homotopy equivalence of diagrams is considered pointwise), then the homotopy limit and colimits then correspond to the cone and cocone which are objects in the homotopy category , where is the category with one object and one morphism. Note this category is equivalent to the standard homotopy category since the latter homotopy functor category has functors which picks out an object in 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.