A comodule-bialgebra structure for word-series substitution and mould composition

Hereditary species as monoidal decomposition spaces, comodule bialgebras, and operadic categories

We show that Schmitt's hereditary species induce monoidal decomposition spaces, and exhibit Schmitt's bialgebra construction as an instance of the general bialgebra construction on a monoidal decomposition space. We show furthermore that this bialgebra structure coacts on the underlying restriction-species bialgebra structure so as to form a comodule bialgebra. Finally, we show that hereditary species induce a new family of examples of operadic categories in the sense of Batanin and Markl.

The incidence comodule bialgebra of the Baez-Dolan construction

Starting from any operad P, one can consider on one hand the free operad on P, and on the other hand the Baez--Dolan construction on P. These two new operads have the same space of operations, but with very different notions of arity and substitution. The main result of this paper is that the incidence bialgebras of the two-sided bar constructions of the two operads constitute together a comodule bialgebra. The result is objective: it concerns comodule-bialgebra structures on groupoid slices, and the proof is given in terms of equivalences of groupoids and homotopy pullbacks. Comodule bialgebr…

Directed hereditary species and decomposition spaces

We introduce the notion of directed hereditary species and show that they have associated monoidal decomposition spaces, comodule bialgebras, and operadic categories. The notion subsumes Schmitt's hereditary species, Gálvez--Kock--Tonks directed restrictions species, and a directed version of Carlier's construction of monoidal decomposition spaces and comodule bialgebras. In addition to all the examples of Schmitt, Gálvez--Kock--Tonks and Carlier, the new construction covers also the Fauvet--Foissy--Manchon comodule bialgebra of finite topological spaces, the Calaque--Ebrahimi-Fard--Manchon co…

On bialgebras and Hopf algebras of oriented graphs

We define two coproducts for cycle-free oriented graphs, thus building up two commutative con- nected graded Hopf algebras, such that one is a comodule-coalgebra on the other, thus generalizing the result obtained previously for Hopf algebras of rooted trees.

Contractions and extractions on twisted bialgebras and coloured Fock functors

We introduce a notion of extraction-contraction coproduct on twisted bialgebras, that is to say bialgebras in the category of linear species. If $P$ is a twisted bialgebra, a contraction-extraction coproduct sends $P[X]$ to $P[X/\sim]\otimes P[X]$ for any finite set $X$ and any equivalence relation $\sim$ on $X$, with a coassociativity constraint and compatibilities with the product and coproduct of $P$. We prove that if $P$ is a twisted bialgebra with an extraction-contraction coproduct, then $P\circ Com$ is a bialgebra in the category of coalgebraic species, that is to say species in the cat…

扩展广义排列多面体与共作用双代数

Extended generalized permutahedra, and cointeracting bialgebras

研究扩展广义排列多面体上共作用双代数结构，通过测量代数框架找到合适共作用概念，证明EGP和仿射锥EGP形成共作用双幺半群，明确其与扩展次模函数对应关系及面和切锥的次模函数，辫子扇等起关键作用。