科研之友
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摘要
Given a bisimplicial set, there are two ways to extract from it a simplicial set: the diagonal simplicial set and the less well known total simplicial set of Artin and Mazur. There is a natural comparison map between these simplicial sets, and it is a theorem due to Cegarra and Remedios and independently Joyal and Tierney, that this comparison map is a weak homotopy equivalence for any bisimplicial set. In this paper we will give a new, elementary proof of this result. As an application, we will revisit Kan%26apos;s simplicial loop group functor G. We will give a simple formula for this functor, which is based on a factorization, due to Duskin, of Eilenberg and Mac Lane%26apos;s classifying complex functor (W) over bar. We will give a new, short, proof of Kan%26apos;s result that the unit map for the adjunction G (sic) (W) over bar is a weak homotopy equivalence for reduced simplicial sets.
- 出版日期2012
