Abstract
AbstractIncidence coalgebras of categories in the sense of Joni and Rota are studied, specifically cases where a monoidal product on the category turns these into (weak) bialgebras. The overlap with the theory of combinatorial Hopf algebras and that of Hopf quivers is discussed, and examples including trees, skew shapes, Milner’s bigraphs and crossed modules are considered.
Highlights
Incidence coalgebras of categories, defined in [18], have been studied in several areas, notably Mobius inversion and combinatorial Hopf algebras
We show that not all Hopf algebras of a combinatorial nature can be described this way, even when the coalgebra structure is the incidence coalgebra of a category
An illustrative, and perhaps counterintuitive, example we investigate the Hopf quivers of Cibils and Rosso: THEOREM 2
Summary
Incidence coalgebras of categories, defined in [18], have been studied in several areas, notably Mobius inversion (see e.g., [21, 22, 23]) and combinatorial Hopf algebras (see e.g., [9, 14, 15, 24]). If a monoidal product on a Mobius category C has the unique lifting of factorisation (ULF) property, its linearisation turns the incidence coalgbera of C into a pointed bialgebra This is a Hopf algebra provided that the monoid of objects is a group. (1) As C is Mobius, the length filtration is an exhaustive coalgebra filtration and C is one-way, so that kC0 is spanned by the monoid (IdC, ·, i1) ∼= (ObC, ·, 1) whose elements are all group-like, (ix) = ix ⊗ ix. This coalgebra is pointed, and by Lemma 4, all simple subcoalgebras of kC are contained in kC0. The remainder of this section is devoted to the discussion of some examples of combinatorial categories and their incidence bialgebras
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
