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58
DG quotients of DG categories
 J. Algebra
"... Abstract. Keller introduced a notion of quotient of a differential graded category modulo a full differential graded subcategory which agrees with Verdier’s notion of quotient of a triangulated category modulo a triangulated subcategory. This work is an attempt to further develop his theory. ..."
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Abstract. Keller introduced a notion of quotient of a differential graded category modulo a full differential graded subcategory which agrees with Verdier’s notion of quotient of a triangulated category modulo a triangulated subcategory. This work is an attempt to further develop his theory.
From subfactors to categories and topology I. Frobenius algebras in and Morita equivalence of tensor categories
 J. Pure Appl. Alg
, 2003
"... We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground f ..."
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Cited by 52 (6 self)
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We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground fields, and the proof of new results about topological invariants in three dimensions. The central notion is that of a Frobenius algebra in a tensor category A, which reduces to the classical notion if A = FVect, where F is a field. An object X ∈ A with twosided dual X gives rise to a Frobenius algebra in A, and under weak additional conditions we prove a converse: There exists a bicategory E with ObjE = {A, B} such that EndE(A) ⊗ ≃ A and such that there are J, J: B ⇋ A producing the given Frobenius algebra. Many properties (additivity, sphericity, semisimplicity,...) of A carry over to the bicategory E. We define weak monoidal Morita equivalence of tensor categories, denoted A ≈ B, and establish a correspondence between Frobenius algebras in A and tensor categories B ≈ A. While considerably weaker than equivalence of tensor categories, weak monoidal Morita equivalence A ≈ B has remarkable consequences: A and B have equivalent (as braided tensor categories) quantum doubles (‘centers’) and (if A, B are semisimple spherical or ∗categories) have equal dimensions and give rise the same state sum invariant of closed oriented 3manifolds as recently defined by Barrett and Westbury. An instructive example is provided by finite dimensional semisimple and cosemisimple Hopf algebras, for which we prove H − mod ≈ ˆH − mod. The present formalism permits a fairly complete analysis of the center of a semisimple spherical category, which is the subject of the companion paper math.CT/0111205. 1
Simplicial Matrices And The Nerves Of Weak nCategories I: Nerves Of Bicategories
, 2002
"... To a bicategory B (in the sense of Benabou) we assign a simplicial set Ner(B), the (geometric) nerve of B, which completely encodes the structure of B as a bicategory. As a simplicial set Ner(B) is a subcomplex of its 2Coskeleton and itself isomorphic to its 3Coskeleton, what we call a 2dimensio ..."
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To a bicategory B (in the sense of Benabou) we assign a simplicial set Ner(B), the (geometric) nerve of B, which completely encodes the structure of B as a bicategory. As a simplicial set Ner(B) is a subcomplex of its 2Coskeleton and itself isomorphic to its 3Coskeleton, what we call a 2dimensional Postnikov complex. We then give, somewhat more delicately, a complete characterization of those simplicial sets which are the nerves of bicategories as certain 2dimensional Postnikov complexes which satisfy certain restricted "exact hornlifting" conditions whose satisfaction is controlled by (and here defines) subsets of (abstractly) invertible 2 and 1simplices. Those complexes which have, at minimum, their degenerate 2simplices always invertible and have an invertible 2simplex # 1 2 (x 12 , x 01 ) present for each "composable pair" (x 12 , , x 01 ) # # 1 2 are exactly the nerves of bicategories. At the other extreme, where all 2 and 1simplices are invertible, are those Kan complexes in which the Kan conditions are satisfied exactly in all dimensions > 2. These are exactly the nerves of bigroupoids  all 2cells are isomorphisms and all 1cells are equivalences. Contents
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 18 (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
Specifying Interaction Categories
, 1997
"... We analyse two complementary methods for obtaining models of typed process calculi, in the form of interaction categories. These methods allow adding new features to previously captured notions of process and of type, respectively. By combining them, all familiar examples of interaction categories, ..."
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We analyse two complementary methods for obtaining models of typed process calculi, in the form of interaction categories. These methods allow adding new features to previously captured notions of process and of type, respectively. By combining them, all familiar examples of interaction categories, as well as some new ones, can be built starting from some simple familiar categories. Using the presented constructions, interaction categories can be analysed without fixing a set of axioms, merely in terms of the way in which they are specified  just like algebras are analysed in terms of equations and relations, independently on abstract characterisations of their varieties.
On PropertyLike Structures
, 1997
"... A category may bear many monoidal structures, but (to within a unique isomorphism) only one structure of "category with finite products". To capture such distinctions, we consider on a 2category those 2monads for which algebra structure is essentially unique if it exists, giving a precise mathemat ..."
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A category may bear many monoidal structures, but (to within a unique isomorphism) only one structure of "category with finite products". To capture such distinctions, we consider on a 2category those 2monads for which algebra structure is essentially unique if it exists, giving a precise mathematical definition of "essentially unique" and investigating its consequences. We call such 2monads propertylike. We further consider the more restricted class of fully propertylike 2monads, consisting of those propertylike 2monads for which all 2cells between (even lax) algebra morphisms are algebra 2cells. The consideration of lax morphisms leads us to a new characterization of those monads, studied by Kock and Zoberlein, for which "structure is adjoint to unit", and which we now call laxidempotent 2monads: both these and their colaxidempotent duals are fully propertylike. We end by showing that (at least for finitary 2monads) the classes of propertylikes, fully propertylike...
Monads And Interpolads In Bicategories
, 1997
"... . Given a bicategory, Y , with stable local coequalizers, we construct a bicategory of monads Y mnd by using lax functors from the generic 0cell, 1cell and 2cell, respectively, into Y . Any lax functor into Y factors through Y mnd and the 1cells turn out to be the familiar bimodules. The local ..."
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Cited by 8 (4 self)
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. Given a bicategory, Y , with stable local coequalizers, we construct a bicategory of monads Y mnd by using lax functors from the generic 0cell, 1cell and 2cell, respectively, into Y . Any lax functor into Y factors through Y mnd and the 1cells turn out to be the familiar bimodules. The locally ordered bicategory rel and its bicategory of monads both fail to be Cauchycomplete, but have a wellknown Cauchycompletion in common. This prompts us to formulate a concept of Cauchycompleteness for bicategories that are not locally ordered and suggests a weakening of the notion of monad. For this purpose, we develop a calculus of general modules between unstructured endo1cells. These behave well with respect to composition, but in general fail to have identities. To overcome this problem, we do not need to impose the full structure of a monad on endo1cells. We show that associative coequalizing multiplications suffice and call the resulting structures interpolads. Together with str...
Maps II: Chasing Diagrams in Categorical Proof Theory
, 1996
"... In categorical proof theory, propositions and proofs are presented as objects and arrows in a category. It thus embodies the strong constructivist paradigms of propositionsastypes and proofsasconstructions, which lie in the foundation of computational logic. Moreover, in the categorical setting, ..."
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Cited by 7 (4 self)
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In categorical proof theory, propositions and proofs are presented as objects and arrows in a category. It thus embodies the strong constructivist paradigms of propositionsastypes and proofsasconstructions, which lie in the foundation of computational logic. Moreover, in the categorical setting, a third paradigm arises, not available elsewhere: logicaloperationsasadjunctions. It offers an answer to the notorious question of the equality of proofs. So we chase diagrams in algebra of proofs. On the basis of these ideas, the present paper investigates proof theory of regular logic: the f; 9gfragment of the first order logic with equality. The corresponding categorical structure is regular fibration. The examples include stable factorisations, sites, triposes. Regular logic is exactly what is needed to talk about maps, as total and singlevalued relations. However, when enriched with proofsasarrows, this familiar concept must be supplied with an additional conversion rule, conn...
Higher cospans and weak cubical categories (Cospans in Algebraic Topology
 I), Theory Appl. Categ
"... form a cubical set with compositions x +i y in all directions, which are computed using pushouts and behave ‘categorically ’ in a weak sense, up to suitable comparisons. Actually, we work with a ‘symmetric cubical structure’, which includes the transposition symmetries, because this allows for a str ..."
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Cited by 5 (2 self)
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form a cubical set with compositions x +i y in all directions, which are computed using pushouts and behave ‘categorically ’ in a weak sense, up to suitable comparisons. Actually, we work with a ‘symmetric cubical structure’, which includes the transposition symmetries, because this allows for a strong simplification of the coherence conditions. These notions will be used in subsequent papers to study topological cospans and their use in Algebraic Topology, from tangles to cobordisms of manifolds. We also introduce the more general notion of a multiple category, where to start witharrows belong to different sorts, varying in a countable family, and symmetries must be dropped. The present examples seem to show that the symmetric cubical case is better suited for topological applications.
Paths in double categories
 Theory Appl. Categ
"... Abstract. Two constructions of paths in double categories are studied, providing algebraic versions of the homotopy groupoid of a space. Universal properties of these constructions are presented. The first is seen as the codomain of the universal oplax morphism of double categories and the second, w ..."
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Abstract. Two constructions of paths in double categories are studied, providing algebraic versions of the homotopy groupoid of a space. Universal properties of these constructions are presented. The first is seen as the codomain of the universal oplax morphism of double categories and the second, which is a quotient of the first, gives the universal normal oplax morphism. Normality forces an equivalence relation on cells, a special case of which was seen before in the free adjoint construction. These constructions are the object part of 2comonads which are shown to be oplax idempotent. The coalgebras for these comonads turn out to be Leinster’s fcmulticategories, with representable identities in the second case.