A simplicial approach to multiplier bimonoids

Abstract

Although multiplier bimonoids in general are not known to correspond to comonoids in any monoidal category, we classify them in terms of maps from the Catalan simplicial set to another suitable simplicial set; thus they can be regarded as (co)monoids in something more general than a monoidal category (namely, the simplicial set itself). We analyze the particular simplicial maps corresponding to that class of multiplier bimonoids which can be regarded as comonoids.

Monads of oplax actions are skew monoidales

“…In [BGLS14], [Buc16], [Gre15], and [BL17] the authors classify various monoidal-like notions as simplicial morphisms out of a particular simplicial set C, whose name is the Catalan simplicial set. One might think of the Catalan simplicial set as the "free living monoidal-like structure" in the sense that one decides which kind of monoidal-like structure to get by choosing different kinds of nerves.…”

A Category of Multiplier Bimonoids

“…Recall that comonoids in a monoidal category M can be regarded as simplicial maps from the Catalan simplicial set C to the nerve of M co (meaning the category with the reverse composition) [5]. In [1] we construct a simplicial set which is not necessarily the nerve of any monoidal category, but for which the simplicial maps from C to it can be identified with multiplier bimonoids.…”

Weak Multiplier Bimonoids

“…Under further assumptions, we constructed a monoidal category of representations of a multiplier bimonoid. We further developed the theory of multiplier bimonoids in two subsequent papers [6,7]. Then in [8] we turned to multiplier Hopf monoids; that is, multiplier bimonoids with a suitable antipode map.…”