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BiHom Hopf algebras viewed as Hopf monoids
- Source :
- Contemporary mathematics-American Mathematical Society
- Publication Year :
- 2020
-
Abstract
- We introduce monoidal categories whose monoidal products of any positive number of factors are lax coherent and whose nullary products are oplax coherent. We call them $\mathsf{Lax}^+\mathsf{Oplax}^0$-monoidal. Dually, we consider $\mathsf{Lax}_0\mathsf{Oplax}_+$-monoidal categories which are oplax coherent for positive numbers of factors and lax coherent for nullary monoidal products. We define $\mathsf{Lax}^+_0\mathsf{Oplax}^0_+$-duoidal categories with compatible $\mathsf{Lax}^+\mathsf{Oplax}^0$- and $\mathsf{Lax}_0\mathsf{Oplax}_+$-monoidal structures. We introduce comonoids in $\mathsf{Lax}^+\mathsf{Oplax}^0$-monoidal categories, monoids in $\mathsf{Lax}_0\mathsf{Oplax}_+$-monoidal categories and bimonoids in $\mathsf{Lax}^+_0\mathsf{Oplax}^0_+$- duoidal categories. Motivation for these notions comes from a generalization of a construction due to Caenepeel and Goyvaerts. This assigns a $\mathsf{Lax}^+_0\mathsf{Oplax}^0_+$-duoidal category $\mathsf D$ to any symmetric monoidal category $\mathsf V$. The unital $\mathsf{BiHom}$-monoids, counital $\mathsf{BiHom}$-comonoids, and unital and counital $\mathsf{BiHom}$-bimonoids in $\mathsf V$ are identified with the monoids, comonoids and bimonoids in $\mathsf D$, respectively.<br />40 pages, a few figures of commutative diagrams
- Subjects :
- Pure mathematics
Unital
Mathematics::General Topology
Symmetric monoidal category
Mathematics - Category Theory
Hopf algebra
Mathématiques
Mathematics::Logic
Mathematics::Probability
Mathematics::Category Theory
Mathematics - Quantum Algebra
FOS: Mathematics
Quantum Algebra (math.QA)
Mathematics::Metric Geometry
Category Theory (math.CT)
Mathematics
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Journal :
- Contemporary mathematics-American Mathematical Society
- Accession number :
- edsair.doi.dedup.....debf34d809149067d0f3294a95163a40