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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 ..."
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Cited by 39 (4 self)
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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 &quot;strongly separable &quot; Frobenius algebras and &quot;weak monoidal Morita equivalence&quot;. 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 ..."
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Cited by 19 (6 self)
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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 ..."
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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
The low dimensional structures that tricategories form
, 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 ..."
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Cited by 6 (1 self)
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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.
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. ..."
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Cited by 4 (1 self)
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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.
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 ..."
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Cited by 2 (1 self)
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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
Twodimensional linear algebra
 Proc. CMCS 2001 ENTCS
, 2001
"... We introduce twodimensional linear algebra, by which we do not mean twodimensional vector spaces but rather the systematic replacement in linear algebra of sets by categories. This entails the study of categories that are simultaneously categories of algebras for a monad and categories of coalgeb ..."
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We introduce twodimensional linear algebra, by which we do not mean twodimensional vector spaces but rather the systematic replacement in linear algebra of sets by categories. This entails the study of categories that are simultaneously categories of algebras for a monad and categories of coalgebras for comonad on a category such as SymMons, the category of small symmetric monoidal categories. We outline relevant notions such as that of pseudoclosed 2category, symmetric monoidal Lawvere theory, and commutativity of a symmetric monoidal Lawvere theory, and we explain the role of coalgebra, explaining its precedence over algebra in this setting. We outline salient results and perspectives given by the dual approach of algebra and coalgebra, extending to two dimensions the study of linear algebra. 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 ..."
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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 ..."
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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.