Commutative classifying space for simplicial groups

Cihan Okay

doi:10.1007/s40062-026-00397-w

Commutative classifying space for simplicial groups

2026-02-01

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DOI：10.1007/s40062-026-00397-w

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In this paper, we introduce a simplicial analog of classifying spaces for commutativity which classify principal bundles with commutativity structure on their transition functions. Our construction W & strns;(tau,K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{W}(\tau ,K)$$\end{document}, which takes as input a simplicial group K and a cosimplicial group tau\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} that encodes the additional structure such as commutativity, is a variation of the W & strns;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{W}$$\end{document}-construction for simplicial groups. Our main result shows that the geometric realization of our W & strns;(tau,K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{W}(\tau ,K)$$\end{document} is homotopy equivalent to the topological classifying space B(tau,|K|)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B(\tau ,|K|)$$\end{document}.

Keywords:

simplicial groups

classifying spaces

commutativity

homotopy equivalence

W-construction

Commutative classifying space for simplicial groups

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