The relationship between the diagonal and the bar constructions on a bisimplicial set. (English) Zbl 1077.55011
There are two simplicial sets associated to a bisimplicial set, the diagonal and the bar construction. The authors show that the two simplicial sets are naturally weakly homotopy equivalent. As applications they prove Thomason’s homotopy colimit theorem for diagrams of small categories and Dold-Puppe’s generalised Eilenberg-Zilber theorem for bisimplicial abelian groups. They also show that the bar construction induces an equivalence between the categories of bisimplicial sets and simplicial sets, give a simplicial model for the classifying space of a \(2\)-category, and give an algebraic model for the homotopy theory of not necessarily connected spaces whose homotopy groups vanish above dimension \(3\).
Reviewer: Richard John Steiner (Glasgow)
MSC:
| 55U10 | Simplicial sets and complexes in algebraic topology |
| 18A30 | Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.) |
| 18D05 | Double categories, \(2\)-categories, bicategories and generalizations (MSC2010) |
| 18G30 | Simplicial sets; simplicial objects in a category (MSC2010) |
Keywords:
bisimplicial set; bar construction; homotopy colimit theorem; Eilenberg-Zilber theorem; homotopy type; 2-categoryReferences:
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