Ring operads and symmetric bimonoidal categories

Kailin Pan
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Abstract

We generalize the classical operad pair theory to a new model for $E_\infty$ ring spaces, which we call ring operad theory, and establish a connection with the classical operad pair theory, allowing the classical multiplicative infinite loop machine to be applied to algebras over any $E_\infty$ ring operad. As an application, we show that classifying spaces of symmetric bimonoidal categories are directly homeomorphic to certain $E_\infty$ ring spaces in the ring operad sense. Consequently, this provides an alternative construction from symmetric bimonoidal categories to classical $E_\infty$ ring spaces. We also present a comparison between this construction and the classical approach.
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环操作数和对称双元范畴
我们将经典操作数对理论推广到一个新的$E_\infty$环空间模型,我们称之为环操作数理论,并建立了与经典操作数对理论的联系,使得经典乘法无限循环机可以应用于任意$E_\infty$环操作数上的代数。作为一个应用,我们证明了对称类的分类空间在环操作数意义上直接同构于某些$E_\infty$环空间。因此,这提供了从对称双元范畴到经典$E_\infty$环空间的另一种构造。我们还比较了这种构造和经典方法。
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