A Category of Multiplier Bimonoids

Abstract

Based on the novel notion of 'weakly counital fusion morphism', regular weak multiplier bimonoids in braided monoidal categories are introduced. They generalize weak multiplier bialgebras over fields [4] and multiplier bimonoids in braided monoidal categories [5]. Under some assumptions the so-called base object of a regular weak multiplier bimonoid is shown to carry a coseparable comonoid structure; hence to possess a monoidal category of bicomodules. In this case, appropriately defined modules over a regular weak multiplier bimonoid are proven to constitute a monoidal category with a strict monoidal forgetful type functor to the category of bicomodules over the base object. Braided monoidal categories considered include various categories of modules or graded modules, the category of complete bornological spaces, and the category of complex Hilbert spaces and continuous linear transformations.

A simplicial approach to multiplier bimonoids

“…Multiplier bimonoids which are comonoids. In our paper [4], following some ideas in [8], we associated to any braided monoidal category C a monoidal category M, and we described a correspondence between certain multiplier bimonoids in C and certain comonoids in M [4, Theorem 5.1]. In this section, we explain this correspondence in terms of simplicial maps and the Catalan simplicial set.…”

“…We now turn to the details. To define the monoidal category M, in [4] we fixed a class Q of regular epimorphisms in C which is closed under composition and monoidal product, contains the isomorphisms, and is right-cancellative in the sense that if s : A → B and t.s : A → C are in Q, then so is t : B → C. Since each q ∈ Q is a regular epimorphism, it is the coequalizer of some pair of morphisms. Finally we suppose that this pair may be chosen in such a way that the coequalizer is preserved by taking the monoidal product with any fixed object.…”

“…Generalizing some constructions in [8] to any braided monoidal category C, we described in [4] how certain multiplier bimonoids in C can be seen as certain comonoids in an appropriately constructed monoidal category M. In light of the findings of [6], this gives rise to a correspondence between these multiplier bialgebras and certain simplicial maps from the Catalan simplicial set C to N(M op ).…”

“…The aim of this paper is to go beyond that characterization and prove a bijection between arbitrary multiplier bimonoids in C and arbitrary simplicial maps from C to a suitable simplicial set M 12 . The simplicial maps C → N(M op ) that correspond, via Date: August 24, 2018. the results of [4] and [6], to nice enough multiplier bimonoids, turn out to factorize through a canonical embedding of a simplicial subset of M 12 into N(M op ).…”

“…If C is a closed braided monoidal category with pullbacks, then for any semigroup B with non-degenerate multiplication one can define its multiplier monoid M(B), see [4]. It is a monoid in C and a universal object characterized by the property that…”