Hopf monoids in the category of species

Abstract

Abstract. A Hopf monoid (in Joyal's category of species) is an algebraic structure akin to that of a Hopf algebra. We provide a self-contained introduction to the theory of Hopf monoids in the category of species. Combinatorial structures which compose and decompose give rise to Hopf monoids. We study several examples of this nature. We emphasize the central role played in the theory by the Tits algebra of set compositions. Its product is tightly knit with the Hopf monoid axioms, and its elements constitute universal operations on connected Hopf monoids. We study analogues of the classical Eulerian and Dynkin idempotents and discuss the Poincaré-Birkhoff-Witt and Cartier-Milnor-Moore theorems for Hopf monoids.

The Hopf Monoid of Hypergraphs and its Sub-Monoids: Basic Invariant and Reciprocity Theorem

“…We recall here basic definitions on Hopf monoids. The interested reader may refer to [4] and to [1] for more information on this topic. In this paper, k is a field and all vector spaces are over k. A sub-monoid of a Hopf monoid M is a sub-species of M stable under the product and coproduct maps.…”

“…. , k l(P ) + 1), (and notice that the set of such choices is empty if l(P ) > n, which allows us not to add this non polynomial dependency in n at the previous choice), 4. choose the colors of the yet uncolored vertices which are in the same edge than a vertex of minimal color in f (H) (k ); then those in the same edge than a vertex of second minimal…”

“…The notion of Hopf algebra is well known and used in combinatorics for over 30 years, and has proved its great strength in various questions (see for example [10]). More recently, Aguiar and Mahajan defined a notion of Hopf monoid [3], [4] akin to the notion of Hopf algebra and built on Joyal's theory of species [12]. Such as in the case of Hopf algebras, a useful application of Hopf monoids is to define and compute polynomial invariants (see [2], [7], [9] or [13] for various examples), as was put to light by the recent and extensive paper of Aguiar and Ardila [1].…”

Lumpings of Algebraic Markov Chains Arise from Subquotients

“…A weak-composition D (also called a decomposition in [AM13]) is a list of non-negative integers d 1 , d 2 , . .…”

Monads on Higher Monoidal Categories

“…A (set) species is a functor set × → Set There is also a distinguished element 1 ∈ B[∅]. The structure is subject to the axioms described in [4,Section 4.2]. While the maps µ and the element 1 turn B into a monoid in the monoidal category (Sp, ·, U), the maps ∆ do not endow it with the structure of a comonoid therein.…”