# 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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