On some Hopf monoids in graphical species

The notion of combinatorial Hopf algebra has been extended to Joyal’s category of species by Aguiar and Mahajan [AM10], so we can now speak of Hopf monoids in species. The idea is that now our combinatorial objects come with labels (from some finite set S), and the objects do not depend on the labels. In this case, the notion of product is to combine combinatorial objects whose label sets are disjoint, and the notion of coproduct is to decompose a combinatorial object into pairs of objects with disjoint label sets.

One can discuss studying other monoidal categories of interest in combinatorics. Bergeron and Choquette [BC10] have studied H-species, which come from hyperoctahedral groups. They also studied various functors, and the resulting Hopf algebras. In some sense, they were looking at combinatorial objects of ’type B’. For the present paper, we are instead interested in graph-theoretic objects.

In some sense, the work of Aguiar and Mahajan [AM10] gives rise to the idea that, instead of attempting to view something as being a combinatorial Hopf algebra, it makes sense to consider what category the combinatorial objects really belong to. For instance, labeled combinatorial objects should be viewed as species. In many cases, labeled combinatorial objects can be generalized to being combinatorial objects on graphs (and usually one recovers the original labeled combinatorial objects by restricting to complete graphs). For instance, linear orders generalize to acyclic orientations, labeled trees generalize to spanning trees, and so on. Hence, there is some motivation to asking whether or not the corresponding ‘graph-theoretic’ analogues also form Hopf monoids.

In this paper, we initialize the study of Hopf monoids in the category of graphical species. That is, now we are studying combinatorial structures on finite graphs, and various ways to combine and decompose them across induced subgraphs. It turns out that many well-known combinatorial structures (matchings, stable partitions, acyclic orientations) carry Hopf monoid structures when viewed as graphical species. Moreover, some of these graphical species come with Hopf monoid morphisms, and the Hopf monoid structures are a ‘graph-theoretic’ analogue of known Hopf monoids in species. This paper will mainly serve for giving definitions of graphical species, as well as detailing several examples. In future papers, we shall discuss other interesting ideas motivated by this paper, and related to graphical species.

The layout is as follows: in the next section we review the definition of the category Sp of species, its lax braided monoidal structure, and several examples of Hopf monoids in species. We also review the notion of bilax monoidal functor. In Section 3, we define the category of graphical species, GrSp, give it a lax braided monoidal category, and define Hopf monoids in GrSp with respect to the braided monoidal structure. In Section 4, we study examples of Hopf monoids related to linear orders, including acyclic orientations and stable compositions. We also relate these Hopf monoids via commutative diagrams. In Section 5, we study examples related to set partitions, including stable partitions and flats. We also relate these Hopf monoids through commutative diagrams. In Section 6, we study how graph complementation gives rise to a bistrong endofunctor on GrSp that changes the braiding map. Then we mention some bilax monoidal functors from GrSp to Sp, which explain why several of the known Hopf monoids in Sp are ‘shadows’ of the Hopf monoids in GrSp we discuss in this paper. Finally, in Section 7, we mention several future directions.

2.1. Species. Throughout this paper, fix a field K. For a definition of lax braided monoidal category, see Joyal and Street [JS93] or Aguiar and Mahajan [AM10]. For the definitions of monoids, comonoids, bimonoids and Hopf monoids in monoidal categories, see Aguiar and Mahajan [AM10]. We will discuss these definitions in the context of the lax braided monoidal category of graphical species with respect to Cauchy product.

Let Set be the category of all finite sets with bijections as morphisms. Let Vec be the category of all vector spaces over K with linear maps as morphisms.

A species is a functor p : Set → Vec. That is, for each finite set I, we associate a vector space p[I], and to each bijection σ : I → J, we associate a linear map

A morphism ϕ : p → q of species is a natural transformation from p to q. That is, for each finite set I we have a linear map ϕ I : p[I] → q[I] such that for any bijection σ : I → J we have the following commutative diagram:

Let

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