\(E_ n\)-monoidal categories and their group completions
Let \(\mathbb{R}_ +\) be the additive topological monoid of nonnegative reals and \(\mathbb{R}^*_ +\) the multiplicative topological group of positive reals, considered as permutative topological categories with all morphisms identities. An \(E_ n\)-monoidal category is a based topological category \({\mathcal A}\) with an \(\mathbb{R}^*_ +\)-action together with continuous \(\mathbb{R}^*_ +\)-equivariant based functors \(\mu_ i : {\mathcal A}^ 2 \to {\mathcal A}\), \(i = 1,2, \dots, n\), such that \(({\mathcal A}, \mu_ i,*)\) is strict monoidal for all \(i\), and \(({\mathcal A}, \mu_ i,*)\) contains \(\mathbb{R}_ +\) as a monoidal retract; moreover, the \(\mu_ i\) satisfy a certain interchange condition. Let \({\mathcal M}_ n {\mathcal C}at\) denote the category of \(E_ n\)-monoidal categories and structure preserving functors. In the first part of the paper the author relates \({\mathcal M}_ n {\mathcal C}at\) to categories of \(C\)-objects in \({\mathcal C}at\), the category of topological categories, where \(C\) is a suitable operad, and of \({\mathbf C}\)-algebras in \({\mathcal C}at\), where \({\mathbf C}\) is a suitable monad. He proves that \({\mathcal M}_ \infty {\mathcal C}at\) is equivalent to the category of topological permutative categories, the equivalences preserving the obvious notions of weak equivalence of these categories. An analogous statement is shown for \({\mathcal M}_ 2 {\mathcal C}at\) and the category of topological braided tensor categories and explained for \({\mathcal M}_ 1 {\mathcal C}at\) and the category of topological strict monoidal categories. In the second part the author defines an \(i\)-fold delooping functor \[ W_ i : {\mathcal M}_ n {\mathcal C}at \to {\mathcal M}_{n - i} {\mathcal C}at \] in the following sense: After inverting weak equivalences there is a natural map \(B {\mathcal A} \to \Omega^ i BW_ i {\mathcal A}\) which is a group completion \((B\) is the classifying space functor). This provides a unified proof of the fact, that after group completion a strictly monoidal category has the homotopy type of a double loop space, and a permutative category the homotopy type of an infinite loop space. The question which purely category theoretical condition on the structure of a tensored category \({\mathcal A}\) makes the group completion of its classifying space an \(n\)-fold loop space for \(n \geq 3\) is not considered.
scientific article; zbMATH DE number 643790