Simplicial and categorical diagrams, and their equivariant applications

Rudolf Fritsch and Marek Golasinski

We show that the homotopy category of simplicial diagrams $I-SS$ indexed by a small category $I$ is equivalent to a homotopy category of $SS\downarrow NI$ simplicial sets over the nerve $NI$. Then their equivalences, by means of the nerve functor N : Cat --> SS$ from the category $Cat$ of small categories, with respective homotopy categories associated to $Cat$ are established. Consequently, an equivariant simplicial version of the Whitehead Theorem is derived.

Keywords: comma category, Grothendieck construction, homotopy colimit, pull-back, simplicial set, small category.

1991 MSC: Primary 55P15, 55U10; secondary 18G30, 55P91.

Theory and Applications of Categories, Vol. 4, 1998, No. , pp 73-81.

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