Incidence categories

Abstract

a b s t r a c tGiven a family F of posets closed under disjoint unions and the operation of taking convex subposets, we construct a category C F called the incidence category of F . This category is ''nearly abelian'' in the sense that all morphisms have kernels/cokernels, and possesses a symmetric monoidal structure akin to direct sum. The Ringel-Hall algebra of C F is isomorphic to the incidence Hopf algebra of the collection P (F ) of order ideals of posets in F . This construction generalizes the categories introduced by K. Kremnizer and the author, in the case when F is the collection of posets coming from rooted forests or Feynman graphs.

Shuffle Algebras and Non-Commutative Probability for Pairs of Faces

“…Remark. The terminology incidence category we use in this paper is not to be confused with occurences in the literature with different meaning, see [Szc11].…”

The Hopf algebra of skew shapes, torsion sheaves on A/F1n, and ideals in Hall algebras of monoid

“…Here, Ob(C) typically consist of combinatorial structures equipped an operation of "collapsing" a sub-structure, which corresponds to forming a quotient in C. Examples of such C include trees, graphs, posets, matroids, semigroup representations on pointed sets, quiver representations in pointed sets etc. (see [18,[26][27][28][29] ). The product in H C , which counts all extensions between two objects, thus amounts to enumerating all combinatorial structures that can be assembled from the two.…”

Representations of Quivers Over F1 and Hall Algebras

“…The details will appear in [26]. (5) The category Rep(Q, F 1 ) shares many structural similarities with incidence categories [24] as well as the category of Feynman graphs [17]. It may be useful to develop the systematics of "F 1 -linear" categories.…”

“…The definition of reflection functors does not seem to go through naively however, since their definition requires one to make sense of the notion of sum e ′′ =i f e of maps in Vect(F 1 ) having a fixed source/target. (3) The category Rep(Q, F 1 ) nil can sometimes be completely characterized by the poset of submodules of a given module, in which case it is equivalent to an incidence category [24]. This leads to a very simple and elementary combinatorial description of the Hall algebra H Q .…”