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34
Representable Multicategories
 Advances in Mathematics
, 2000
"... We introduce the notion of representable multicategory , which stands in the same relation to that of monoidal category as bration does to contravariant pseudofunctor (into Cat). We give an abstract reformulation of multicategories as monads in a suitable Kleisli bicategory of spans. We describe ..."
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Cited by 34 (6 self)
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We introduce the notion of representable multicategory , which stands in the same relation to that of monoidal category as bration does to contravariant pseudofunctor (into Cat). We give an abstract reformulation of multicategories as monads in a suitable Kleisli bicategory of spans. We describe representability in elementary terms via universal arrows . We also give a doctrinal characterisation of representability based on a fundamental monadic adjunction between the 2category of multicategories and that of strict monoidal categories. The first main result is the coherence theorem for representable multicategories, asserting their equivalence to strict ones, which we establish via a new technique based on the above doctrinal characterisation. The other main result is a 2equivalence between the 2category of representable multicategories and that of monoidal categories and strong monoidal functors. This correspondence extends smoothly to one between bicategories and a se...
Pseudo limits, biadjoints, and pseudo algebras: categorical foundations of conformal field theory
 Mem. Amer. Math. Soc
"... The purpose of this paper is to work out the categorical basis for the foundations of Conformal Field Theory. The definition of Conformal Field Theory was outlined in Segal [45] and recently given in [24] and [25]. Concepts of 2category theory, such as versions of algebra, limit, colimit, and adjun ..."
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Cited by 19 (8 self)
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The purpose of this paper is to work out the categorical basis for the foundations of Conformal Field Theory. The definition of Conformal Field Theory was outlined in Segal [45] and recently given in [24] and [25]. Concepts of 2category theory, such as versions of algebra, limit, colimit, and adjunction, are necessary for this
Theory and Applications of Crossed Complexes
, 1993
"... ... L are simplicial sets, then there is a strong deformation retraction of the fundamental crossed complex of the cartesian product K \Theta L onto the tensor product of the fundamental crossed complexes of K and L. This satisfies various sideconditions and associativity/interchange laws, as for t ..."
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Cited by 15 (2 self)
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... L are simplicial sets, then there is a strong deformation retraction of the fundamental crossed complex of the cartesian product K \Theta L onto the tensor product of the fundamental crossed complexes of K and L. This satisfies various sideconditions and associativity/interchange laws, as for the chain complex version. Given simplicial sets K 0 ; : : : ; K r , we discuss the rcube of homotopies induced on (K 0 \Theta : : : \Theta K r ) and show these form a coherent system. We introduce a definition of a double crossed complex, and of the associated total (or codiagonal) crossed complex. We introduce a definition of homotopy colimits of diagrams of crossed complexes. We show that the homotopy colimit of crossed complexes can be expressed as the
Pasting diagrams in ncategories with applications to coherence theorems and categories of paths
, 1988
"... ..."
Higherdimensional categories with finite derivation type
"... We study convergent (terminating and confluent) presentations of ncategories. Using the notion of polygraph (or computad), we introduce the homotopical property of finite derivation type for ncategories, generalising the one introduced by Squier for word rewriting systems. We characterise this pr ..."
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Cited by 12 (3 self)
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We study convergent (terminating and confluent) presentations of ncategories. Using the notion of polygraph (or computad), we introduce the homotopical property of finite derivation type for ncategories, generalising the one introduced by Squier for word rewriting systems. We characterise this property by using the notion of critical branching. In particular, we define sufficient conditions for an ncategory to have finite derivation type. Through examples, we present several techniques based on derivations of 2categories to study convergent presentations by 3polygraphs.
Frobenius Algebras and ambidextrous adjunctions
, 2006
"... In this paper we explain the relationship between Frobenius objects in monoidal categories and adjunctions in 2categories. Specifically, we show that every Frobenius object in a monoidal category M arises from an ambijunction (simultaneous left and right adjoints) in some 2categoryDinto which M fu ..."
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Cited by 12 (1 self)
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In this paper we explain the relationship between Frobenius objects in monoidal categories and adjunctions in 2categories. Specifically, we show that every Frobenius object in a monoidal category M arises from an ambijunction (simultaneous left and right adjoints) in some 2categoryDinto which M fully and faithfully embeds. Since a 2D topological quantum field theory is equivalent to a commutative Frobenius algebra, this result also shows that every 2D TQFT is obtained from an ambijunction in some 2category. Our theorem is proved by extending the theory of adjoint monads to the context of an arbitrary 2category and utilizing the free completion under EilenbergMoore objects. We then categorify this theorem by replacing the monoidal category M with a semistrict monoidal 2category M, and replacing the 2categoryD into which it embeds by a semistrict 3category. To state this more powerful result, we must first define the notion of a ‘Frobenius pseudomonoid’, which categorifies that of a Frobenius object. We then define the notion of a ‘pseudo ambijunction’, categorifying that of an ambijunction. In each case, the idea is that all the usual axioms now hold only up to coherent isomorphism. Finally, we show that every Frobenius pseudomonoid in a semistrict monoidal 2category arises from a pseudo ambijunction in some semistrict 3category.
Computads for Finitary Monads on Globular Sets
, 1998
"... . A finitary monad A on the category of globular sets provides basic algebraic operations from which more involved `pasting' operations can be derived. To makes this rigorous, we define Acomputads and construct a monad on the category of Acomputads whose algebras are Aalgebras; an action of ..."
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Cited by 10 (1 self)
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. A finitary monad A on the category of globular sets provides basic algebraic operations from which more involved `pasting' operations can be derived. To makes this rigorous, we define Acomputads and construct a monad on the category of Acomputads whose algebras are Aalgebras; an action of the new monad encapsulates the pasting operations. When A is the monad whose algebras are ncategories, an Acomputad is an ncomputad in the sense of R.Street. When A is associated to a higher operad (in the sense of the author) , we obtain pasting in weak ncategories. This is intended as a first step towards proving the equivalence of the various definitions of weak ncategory now in the literature. Introduction This work arose as a reflection on the foundation of higher dimensional category theory. One of the main ingredients of any proposed definition of weak ncategory is the shape of diagrams (pasting scheme) we accept to be composable. In a globular approach [3] each kcell has a source ...
Functorial boxes in string diagrams
, 2006
"... String diagrams were introduced by Roger Penrose as a handy notation to manipulate morphisms in a monoidal category. In principle, this graphical notation should encompass the various pictorial systems introduced in prooftheory (like JeanYves Girard’s proofnets) and in concurrency theory (like Ro ..."
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Cited by 8 (2 self)
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String diagrams were introduced by Roger Penrose as a handy notation to manipulate morphisms in a monoidal category. In principle, this graphical notation should encompass the various pictorial systems introduced in prooftheory (like JeanYves Girard’s proofnets) and in concurrency theory (like Robin Milner’s bigraphs). This is not the case however, at least because string diagrams do not accomodate boxes — a key ingredient in these pictorial systems. In this short tutorial, based on our accidental rediscovery of an idea by Robin Cockett and Robert Seely, we explain how string diagrams may be extended with a notion of functorial box to depict a functor separating an inside world (its source category) from an outside world (its target category). We expose two elementary applications of the notation: first, we characterize graphically when a faithful balanced monoidal functor F: C − → D transports a trace operator from the category D