Colocalizations and their realizations as spectra.

Bauer, Friedrich W.

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Effective homology for homotopy colimit and cofibrant replacement

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We extend the notion of simplicial set with effective homology presented in [22] to diagrams of simplicial sets. Further, for a given finite diagram of simplicial sets $X : ℐ \to sSet$ such that each simplicial set $X (i)$ has effective homology, we present an algorithm computing the homotopy colimit $hocolim X$ as a simplicial set with effective homology. We also give an algorithm computing the cofibrant replacement $X^{cof}$ of $X$ as a diagram with effective homology. This is applied to computing of equivariant cohomology...

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Following a Bendersky-Gitler idea, we construct an isomorphism between Anderson’s and Arone’s complexes modelling the chain complex of a mapping space. This allows us to apply Shipley’s convergence theorem to Arone’s model. As a corollary, we reduce the problem of homotopy equivalence for certain “toy” spaces to a problem in homological algebra.