<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\overline{W}(\tau ,K)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mi>W</mi> <mo>¯</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>,</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, which takes as input a simplicial group <i>K</i> and a cosimplicial group <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\tau $</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation> that encodes the additional structure such as commutativity, is a variation of the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\overline{W}$</EquationSource> <EquationSource Format="MATHML"><math> <mover> <mi>W</mi> <mo>¯</mo> </mover> </math></EquationSource> </InlineEquation>-construction for simplicial groups. Our main result shows that the geometric realization of our <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\overline{W}(\tau ,K)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mi>W</mi> <mo>¯</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>,</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is homotopy equivalent to the topological classifying space <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$B(\tau ,|K|)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>,</mo> <mo stretchy="false">|</mo> <mi>K</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

Commutative classifying space for simplicial groups

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2026年2月19日

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Cihan Okay,
Pál Zsámboki

摘要

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 $\overline{W}(\tau ,K)$ $\bar{W} (τ, K)$ , which takes as input a simplicial group K and a cosimplicial group $\tau $ $τ$ that encodes the additional structure such as commutativity, is a variation of the $\overline{W}$ $\bar{W}$ -construction for simplicial groups. Our main result shows that the geometric realization of our $\overline{W}(\tau ,K)$ $\bar{W} (τ, K)$ is homotopy equivalent to the topological classifying space $B(\tau ,|K|)$ $B (τ, | K |)$ .