Categorifying equivariant monoids (Q6583745)

The idea of combining the theory of PROPs and the theory of crossed simplicial groups to categorify categories of monoids goes back to \textit{T. Pirashvili} [Cah. Topol. Géom. Différ. Catég. 43, No. 3, 221--239 (2002; Zbl 1057.18005)], in which the structure of the symmetric crossed simplicial group is used to construct PROPs whose categories of algebras are monoids, comonoids and bimonoids in a symmetric monoidal category. These results provided a generalization of previous results of \textit{M. Markl} [J. Pure Appl. Algebra 113, No. 2, 195--218 (1996; Zbl 0865.18011)], who constructed such PROPs in terms of generators and relations in order to study the deformation theory of algebras. \textit{S. Lack} [Theory Appl. Categ. 13, 147--163 (2004; Zbl 1062.18007)] showed that Pirashvili(s result fits into a broader categorical framework, namely, that of composing PROPs via a distributive law.\N\NUsing the techniques of Pirashvili and Lack, the author [Theory Appl. Categ. 35, 1564--1575 (2020; Zbl 1455.16030)] extended the existing theory to construct PROPs for monoids, comonoids and bimonoids rigged with an order-preserving involution, using the hyperoctahedral crossed simplicial group. The author [Theory Appl. Categ. 38, 1050--1061 (2022; Zbl 1498.18022)] has also shown that one can extend the results of Lack's paper to the setting of braided monoidal categories, establishing analogues of Pirashvili's result by use of the braid crossed simplicial group, where the categories considered are called PROBs. \textit{M. Kapranov} and \textit{V. Schechtman} [``PROBs and perverse sheaves I. Symmetric products'', Preprint, \url{arXiv:2102.13321}; ``PROBs and perverse sheaves II. Ran spaces and 0-cycles with coefficients'', Preprint, \url{arXiv:2209.02400}] have investigated PROBs and perverse sheaves.\N\NThis paper investigates equivariant extensions of the symmetric, braided and hyperoctahedral crossed simplicial groups, completely generalizing the existing results. In particular, Theorems 5.3, 6.16 and 7.1 specialize to the theorems [\textit{D. Graves}, Theory Appl. Categ. 38, 1050--1061 (2022; Zbl 1498.18022), Theorem 7.2], [\textit{T. Pirashvili}, Cah. Topol. Géom. Différ. Catég. 43, No. 3, 221--239 (2002; Zbl 1057.18005), Theorem 5.2] and [\textit{D. Graves}, Theory Appl. Categ. 35, 1564--1575 (2020; Zbl 1455.16030), Theorem 5.3] respectively. The author also recalls what it means for a braided monoidal category to be balanced, extending the results to this setting by consideration on the ribbon braid crossed simplicial groups and its equivariant extensions.\N\NThe synopsis of the paper goes as follows.\N\N\begin{itemize}\N\item[\S 1] recalls the definitions of \(G\)-monoids, \(G\)-comonoids and \(G\)-bimonoids in a symmetric or braided monoidal category, together with examples of interest from topology and homological algebra.\N\N\item[\S 2] recalls the definitions of PROPs and PROBs as well as the definition of an algebra for these categories in a symmetric or braided monoidal category, providing a list of examples.\N\N\item[\S 3] describes the procedure for forming new PROPs and PROBs via a distributive law in two special cases, showing that the group compositions in certain families of semi-direct products induces a distributive law between the associated groupoids. It is also recalled how the composition law in a crossed simplicial group determines a distributive law.\N\N\item[\S 4] establishes that PROPs and PROBs whose categories of algebras are equivalent to the categories of \(G\)-monoids and \(G\)-comonoids have been constructed for symmetric monoidal categories, braided monoidal categories and balanced braided monoidal categories. The author constructs a PROP for \(G\)-commutative monoids and \(G\)-cocommutative comonoids in a symmetric monoidal category.\N\N\item[\S 5] constructs the PROPs for \(G\)-bimonoids in braided and balanced braided monoidal categories by composing the \(G\)-bimonoids in a symmetric monoidal category.\N\N\item[\S 6] proves similar results for \(G\)-bimonoids in a symmetric monoidal category, constructing the necessary PROPs in two different ways.\N\N\item[\S 7] notes that in a symmetric monoidal category one has monoids that possess both an involution and a group action in a compatible way. The author uses the equivariant extensions of the hyperoctahedral crossed simplicial group to construct a PROP encoding this data. \end{itemize}