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Covariant fibrations and diagrams of spaces

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If you have a question about this talk, please contact Dr Ignacio Lopez Franco.

For a small category A, I consider the category sSetsA of diagrams of simplicial sets (“spaces”) parametrized by A. The usual homotopy colimit functor construction can be considered as a functor

h!: sSets A—> sSets/NA,

where NA is the nerve of A. It is well known that this functor gives an equivalence of homotopy categories when A is group (viewed as a category with one object). I will show that h! always gives an equivalence of homotopy categories, in the following precise way: One equips sSets^A with the projective model structure, and sSets/NA with the covariant model structure. The talk is based on joint work with Gijs Heuts, and simplifies the treatment in Lurie’s Higher Topos Theory.

This talk is part of the Category Theory Seminar series.

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