Results 1 
9 of
9
Frobenius monads and pseudomonoids
 2CATEGORIES COMPANION 73
, 2004
"... Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalenc ..."
Abstract

Cited by 19 (4 self)
 Add to MetaCart
Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalence is defined; this concept is related to "strongly separable " Frobenius algebras and "weak monoidal Morita equivalence". Wreath products of Frobenius algebras are discussed.
Convex hull realizations of the multiplihedra
, 2007
"... Abstract. We present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the n th polytope in the sequence known as the multiplihedra. This answers the open question of whether the multiplihedra could be realized as convex polytopes. Contents ..."
Abstract

Cited by 13 (4 self)
 Add to MetaCart
Abstract. We present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the n th polytope in the sequence known as the multiplihedra. This answers the open question of whether the multiplihedra could be realized as convex polytopes. Contents
Quotients of the multiplihedron as categorified associahedra
 Homotopy, Homology and Appl
, 2008
"... Abstract. We describe a new sequence of polytopes which characterize A ∞ maps from a topological monoid to an A ∞ space. Therefore each of these polytopes is a quotient of the corresponding multiplihedron. Our sequence of polytopes is demonstrated not to be combinatorially equivalent to the associah ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
Abstract. We describe a new sequence of polytopes which characterize A ∞ maps from a topological monoid to an A ∞ space. Therefore each of these polytopes is a quotient of the corresponding multiplihedron. Our sequence of polytopes is demonstrated not to be combinatorially equivalent to the associahedra, as was previously assumed in both topological and categorical literature. They are given the new collective name composihedra. We point out how these polytopes are used to parameterize compositions in the formulation of the theories of enriched bicategories and pseudomonoids in a monoidal bicategory. We also present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the nth polytope in the sequence of
An Australian conspectus of higher categories

, 2004
"... Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higherdimensional wo ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higherdimensional work. I trust that the somewhat autobiographical style will add interest rather than be a distraction. For so long I have felt rather apologetic when describing how categories might be helpful to other mathematicians; I have often felt even worse when mentioning enriched and higher categories to category theorists. This is not to say that I have doubted the value of our work, rather that I have felt slowed down by the continual pressure to defend it. At last, at this meeting, I feel justified in speaking freely amongst motivated researchers who know the need for the subject is well established. Australian Category Theory has its roots in homology theory: more precisely, in the treatment of the cohomology ring and the Künneth formulas in the book by Hilton and Wylie [HW]. The first edition of the book had a mistake concerning the cohomology ring of a product. The Künneth formulas arise from splittings of the natural short exact sequences
The low dimensional structures that tricategories form, preprint http://arxiv.org/abs/0711.1761
, 2007
"... We form tricategories and the homomorphisms between them into a bicategory. We then enrich this bicategory into an example of a threedimensional structure called a locally double bicategory, this being a bicategory enriched in the monoidal 2category of weak double categories. Finally, we show that ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
We form tricategories and the homomorphisms between them into a bicategory. We then enrich this bicategory into an example of a threedimensional structure called a locally double bicategory, this being a bicategory enriched in the monoidal 2category of weak double categories. Finally, we show that every sufficiently wellbehaved locally double bicategory gives rise to a tricategory, and thereby deduce the existence of a tricategory of tricategories. 1
The Compact Closed Bicategory of Left Adjoints
, 1999
"... We show that, for any braided compact closed bicategory B, the bicategory Ladj(B) of left adjoints in B also admits a braided compact closed structure. 1 ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
We show that, for any braided compact closed bicategory B, the bicategory Ladj(B) of left adjoints in B also admits a braided compact closed structure. 1
STRICTIFICATION OF CATEGORIES WEAKLY ENRICHED IN SYMMETRIC MONOIDAL CATEGORIES
"... Abstract. We show that categories weakly enriched over symmetric monoidal categories can be strictified to categories enriched in permutative categories. This is a “many 0cells ” version of the strictification of bimonoidal categories to strict ones. 1. ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. We show that categories weakly enriched over symmetric monoidal categories can be strictified to categories enriched in permutative categories. This is a “many 0cells ” version of the strictification of bimonoidal categories to strict ones. 1.
Example
, 2004
"... Recall the ordinary notion of Frobenius algebra over a field k. Step 2 Lift the concept from linear algebra to a general monoidal category and justify this with examples and theorems. Step 3 Lift the concept up a dimension so that monoidal categories themselves can be examples. 1 Frobenius algebras ..."
Abstract
 Add to MetaCart
Recall the ordinary notion of Frobenius algebra over a field k. Step 2 Lift the concept from linear algebra to a general monoidal category and justify this with examples and theorems. Step 3 Lift the concept up a dimension so that monoidal categories themselves can be examples. 1 Frobenius algebras An algebra A over a field k is called Frobenius when it is finite dimensional and equipped with a linear function e:A æÆ æ k such that: e ( ab) = 0 for all a ŒA implies b = 0.
Openclosed strings: Twodimensional extended TQFTs and
, 2005
"... We study a special sort of 2dimensional extended Topological Quantum Field Theories (TQFTs) which we call openclosed TQFTs. These are defined on openclosed cobordisms by which we mean smooth compact oriented 2manifolds with corners that have a particular global structure in order to model the sm ..."
Abstract
 Add to MetaCart
We study a special sort of 2dimensional extended Topological Quantum Field Theories (TQFTs) which we call openclosed TQFTs. These are defined on openclosed cobordisms by which we mean smooth compact oriented 2manifolds with corners that have a particular global structure in order to model the smooth topology of open and closed string worldsheets.