Stefan Forcey

Marked tubes and the graph multiplihedron

Given a graph G , we construct a convex polytope whose face poset is based on marked subgraphs of G . Dubbed the graph multiplihedron, we provide a realization using integer coordinates. Not only does this yield a natural generalization of the multiplihedron, but features of this polytope appear in works related to quilted disks, bordered Riemann surfaces and operadic structures. Certain examples of graph multiplihedra are related to Minkowski sums of simplices and cubes and others to the permutohedron. 52B11; 18D50, 55P48

Enrichment over iterated monoidal categories

Joyal and Street note in their paper on braided monoidal cate- gories (9) that the 2-category V-Cat of categories enriched over a braided monoidal category V is not itself braided in any way that is based upon the braiding of V. The exception that they mention is the case in which V is symmetric, which leads to V-Cat being symmetric as well. The symmetry in V-Cat is based upon the symmetry of V. The motivation behind this paper is in part to describe how these facts relating V and V-Cat are in turn related to a categorical analogue of topological delooping. To do so I need to pass to a more general setting than braided and symmetric cat- egories — in fact the k-fold monoidal categories of Balteanu et al in (2). It seems that the analogy of loop spaces is a good guide for how to define the concept of enrichment over various types of monoidal objects, including k-fold monoidal categories and their higher dimensional counterparts. The main result is that for V a k-fold monoidal category, V-Cat becomes a (k − 1)-fold monoidal 2-category in a canonical way. In the next paper I indicate how this process may be iterated by enriching over V-Cat, along the way defining the 3-category of categories enriched over V-Cat. In fu- ture work I plan to make precise the n-dimensional case and to show how the group completion of the nerve of V is related to the loop space of the group completion of the nerve of V-Cat. AMS Classification 18D10; 18D20

Geometric combinatorial algebras: cyclohedron and simplex

In this paper we report on results of our investigation into the algebraic structure supported by the combinatorial geometry of the cyclohedron. Our new graded algebra structures lie between two well known Hopf algebras: the Malvenuto–Reutenauer algebra of permutations and the Loday–Ronco algebra of binary trees. Connecting algebra maps arise from a new generalization of the Tonks projection from the permutohedron to the associahedron, which we discover via the viewpoint of the graph associahedra of Carr and Devadoss. At the same time, that viewpoint allows exciting geometrical insights into the multiplicative structure of the algebras involved. Extending the Tonks projection also reveals a new graded algebra structure on the simplices. Finally this latter is extended to a new graded Hopf algebra with basis all the faces of the simplices.

Pseudograph associahedra

Given a simple graph G, the graph associahedron KG is a simple polytope whose face poset is based on the connected subgraphs of G. This paper defines and constructs graph associahedra in a general context, for pseudographs with loops and multiple edges, which are also allowed to be disconnected. We then consider deformations of pseudograph associahedra as their underlying graphs are altered by edge contractions and edge deletions.

Hopf Structures on the Multiplihedra

We investigate algebraic structures that can be placed on vertices of the multiplihedra, a family of polytopes originating in the study of higher categories and homotopy theory. Most compelling among these are two distinct structures of a Hopf module over the Loday-Ronco Hopf algebra.

Convex polytopes from nested posets

Motivated by the graph associahedron K G , a polytope whose face poset is based on connected subgraphs of G , we consider the notion of associativity and tubes on posets. This leads to a new family of simple convex polytopes obtained by iterated truncations. These generalize graph associahedra and nestohedra, even encompassing notions of nestings on CW-complexes. However, these poset associahedra fall in a different category altogether than generalized permutohedra.

Extending the Tamari Lattice to Some Compositions of Species

An extension of the Tamari lattice to the multiplihedra is discussed, along with projections to the composihedra and the Boolean lattice. The multiplihedra and composihedra are sequences of polytopes that arose in algebraic topology and category theory. Here we describe them in terms of the composition of combinatorial species. We define lattice structures on their vertices, indexed by painted trees, which are extensions of the Tamari lattice and projections of the weak order on the permutations. The projections from the weak order to the Tamari lattice and the Boolean lattice are shown to be different from the classical ones. We review how lattice structures often interact with the Hopf algebra structures, following Aguiar and Sottile who discovered the applications of Mobius inversion on the Tamari lattice to the Loday-Ronco Hopf algebra.

Split-Facets for Balanced Minimal Evolution Polytopes and the Permutoassociahedron

Understanding the face structure of the balanced minimal evolution (BME) polytope, especially its top-dimensional facets, is a fundamental problem in phylogenetic theory. We show that BME polytope has a sublattice of its poset of faces which is isomorphic to a quotient of the well-studied permutoassociahedron. This sublattice corresponds to compatible sets of splits displayed by phylogenetic trees and extends the lattice of faces of the BME polytope found by Hodge, Haws and Yoshida. Each of the maximal elements in our new poset of faces corresponds to a single split of the leaves. Nearly all of these turn out to actually be facets of the BME polytope, a collection of facets which grows exponentially.

Classification of braids which give rise to interchange

It is well known that the existence of a braiding in a monoidal category V allows many higher structures to be built upon that foundation. These include a monoidal 2‐category V ‐Cat of enriched categories and functors over V , a monoidal bicategory V ‐Mod of enriched categories and modules, a category of operads in V and a 2‐fold monoidal category structure on V . These all rely on the braiding to provide the existence of an interchange morphism necessary for either their structure or its properties. We ask, given a braiding on V , what non-equal structures of a given kind from this list exist which are based upon the braiding. For example, what non-equal monoidal structures are available on V ‐Cat, or what non-equal operad structures are available which base their associative structure on the braiding in V . The basic question is the same as asking what non-equal 2‐fold monoidal structures exist on a given braided category. The main results are that the possible 2‐fold monoidal structures are classified by a particular set of four strand braids which we completely characterize, and that these 2‐fold monoidal categories are divided into two equivalence classes by the relation of 2‐fold monoidal equivalence. 57M99