Commutative classifying space for simplicial groups

Abstract 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 $$\overline{W}(\tau ,K)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>W</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>K</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , which takes as input a simplicial group K and a cosimplicial group $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> that encodes the additional structure such as commutativity, is a variation of the $$\overline{W}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>W</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> -construction for simplicial groups. Our main result shows that the geometric realization of our $$\overline{W}(\tau ,K)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>W</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>K</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is homotopy equivalent to the topological classifying space $$B(\tau ,|K|)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>(</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>,</mml:mo> <mml:mo>|</mml:mo> <mml:mi>K</mml:mi> <mml:mo>|</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> .