Related papers: Coherence for braided and symmetric pseudomonoids
It is well known that braided monoidal categories are the categorical algebras of the little two-dimensional disks operad. We introduce involutive little disks operads, which are Z/2Z-orbifold versions of the little disks operads. We…
In connection with the emerging theory of Garside categories, we develop the notions of a left-Garside category and of a locally left-Garside monoid. In this framework, the connection between the self-distributivity law LD and braids…
The notion of a braided chord diagram is introduced and studied. An equivalence relation is given which identifies all braidings of a fixed chord diagram. It is shown that finite-type invariants are stratified by braid index for knots which…
A large class of positive finite presentations of the braid groups is found and studied. It is shown that no presentations but known exceptions in this class have the property that equivalent braid words are also equivalent under positive…
We prove that the homotopy theory of parsummable categories (as defined by Schwede) with respect to the underlying equivalences of categories is equivalent to the usual homotopy theory of symmetric monoidal categories. In particular, this…
This paper is addressed to logicians not familiar with category theory. It gives a new proof of coherence for symmetric monoidal closed categories, proven by Kelly and Mac Lane in early 1970s. We find this result of great importance for…
The paper is devoted to prove a version of Milnor-Moore Theorem for connected braided bialgebras that are infinitesimally cocommutative. Namely in characteristic different from 2, we prove that, for a given connected braided bialgebra $A$…
Hopf (bi-)modules and crossed modules over a bialgebra B in a braided monoidal category C are considered. The (braided) monoidal equivalence of both categories is proved provided B is a Hopf algebra (with invertible antipode). Bialgebra…
We give a short topological proof of coherence for categorified non-symmetric operads by using the fact that the diagrams involved form the 1-skeleton of simply connected CW complexes. We also obtain a "one-step" topological proof of Mac…
We describe a perfect correspondence between skew monoidal categories and certain generalised multicategories, called skew multicategories, that arise in nature.
Poly-bicategories generalise planar polycategories in the same way as bicategories generalise monoidal categories. In a poly-bicategory, the existence of enough 2-cells satisfying certain universal properties (representability) induces…
We show that every internal biequivalence in a tricategory T is part of a biadjoint biequivalence. We give two applications of this result, one for transporting monoidal structures and one for equipping a monoidal bicategory with invertible…
Bicategories of spans are characterized as cartesian bicategories in which every comonad has an Eilenberg-Moore ob ject and every left adjoint arrow is comonadic.
We show how monoidal adjunctions can be used to prove the existence of monoidal abelian envelopes of pseudo-tensor categories, in particular, those admitting a combinatorial description with certain properties. We derive concrete general…
We introduce bimonads in a 2-category $\K$ and define biwreaths as bimonads in the 2-category $\bEM(\K)$ of bimonads, in the analogous fashion as Lack and Street defined wreaths. A biwreath is then a system containing a wreath, a cowreath…
Proofs of coherence in category theory, starting from Mac Lane's original proof of coherence for monoidal categories, are sometimes based on confluence techniques analogous to what one finds in the lambda calculus, or in term-rewriting…
We extend the arithmetic product of species of structures and symmetric sequences studied by Maia and Mendez and by Dwyer and Hess to coloured symmetric sequences and show that it determines a normal oplax monoidal structure on the…
Much of algebra and representation theory can be formulated in the general framework of tensor categories. The aim of this paper is to further develop this theory for braided tensor categories. Several results are established that do not…
We define a monoid structure on the set of $k$-equal arrangements and use this structure to define limits of braid arrangements. We compute the cohomology of the associated limits of rational models of the arrangements complex complements.…
In the paper we give a survey on braid groups and subjects connected with them. We start with the initial definition, then we give several interpretations as well as several presentations of these groups. Burau presentation for the pure…