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Operads In HigherDimensional Category Theory
, 2004
"... The purpose of this paper is to set up a theory of generalized operads and multicategories and to use it as a language in which to propose a definition of weak ncategory. Included is a full explanation of why the proposed definition of ncategory is a reasonable one, and of what happens when n < ..."
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The purpose of this paper is to set up a theory of generalized operads and multicategories and to use it as a language in which to propose a definition of weak ncategory. Included is a full explanation of why the proposed definition of ncategory is a reasonable one, and of what happens when n <= 2. Generalized operads and multicategories play other parts in higherdimensional algebra too, some of which are outlined here: for instance, they can be used to simplify the opetopic approach to ncategories expounded by Baez, Dolan and others, and are a natural language in which to discuss enrichment of categorical structures.
Basic bicategories
 Eprint math.CT/9810017
, 1998
"... A concise guide to very basic bicategory theory, from the definition of a bicategory to the coherence theorem. ..."
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Cited by 13 (1 self)
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A concise guide to very basic bicategory theory, from the definition of a bicategory to the coherence theorem.
Framed Bicategories and Monoidal Fibrations
, 2007
"... Abstract. In some bicategories, the 1cells are ‘morphisms ’ between the 0cells, such as functors between categories, but in others they are ‘objects ’ over the 0cells, such ..."
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Abstract. In some bicategories, the 1cells are ‘morphisms ’ between the 0cells, such as functors between categories, but in others they are ‘objects ’ over the 0cells, such
Internal categories, anafunctors and localisations
"... In this article we review the theory of anafunctors introduced by Makkai and Bartels, and show that given a subcanonical site S, one can form a bicategorical localisation of various 2categories of internal categories or groupoids at weak equivalences using anafunctors as 1arrows. This unifies a nu ..."
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In this article we review the theory of anafunctors introduced by Makkai and Bartels, and show that given a subcanonical site S, one can form a bicategorical localisation of various 2categories of internal categories or groupoids at weak equivalences using anafunctors as 1arrows. This unifies a number of proofs throughout the literature, using the fewest assumptions possible on S. 1.
TWOSIDED DISCRETE FIBRATIONS IN 2CATEGORIES AND BICATEGORIES
"... variation models functors B op ×A → Set. By work of Street, both notions can be defined internally to an arbitrary 2category or bicategory. While the twosided discrete fibrations model profunctors internally to Cat, unexpectedly, the dual twosided codiscrete cofibrations are necessary to model Vp ..."
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variation models functors B op ×A → Set. By work of Street, both notions can be defined internally to an arbitrary 2category or bicategory. While the twosided discrete fibrations model profunctors internally to Cat, unexpectedly, the dual twosided codiscrete cofibrations are necessary to model Vprofunctors internally to VCat. There are many categorical prerequisites, particularly in the later sections, but we believe they are strictly easier than the topics below that take advantage of them. These notes were written to accompany a talk given in the Algebraic Topology and Category Theory Proseminar in the fall of 2010 at the University of Chicago.
Abstract
, 2006
"... We describe a Catvalued nerve of bicategories, which associates to every bicategory a simplicial object in Cat, called the 2nerve. This becomes the object part of a 2functor N: NHom → [ ∆ op,Cat], where NHom is a 2category whose objects are bicategories and whose 1cells are normal homomorphisms ..."
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We describe a Catvalued nerve of bicategories, which associates to every bicategory a simplicial object in Cat, called the 2nerve. This becomes the object part of a 2functor N: NHom → [ ∆ op,Cat], where NHom is a 2category whose objects are bicategories and whose 1cells are normal homomorphisms of bicategories. The 2functor N is fully faithful and has a left biadjoint, and we characterize its image. The 2nerve of a bicategory is always a weak 2category in the sense of Tamsamani, and we show that NHom is biequivalent to a certain 2category whose objects are Tamsamani weak 2categories. This paper concerns a notion of “2nerve”, or Catvalued nerve, of bicategories. To every category, one can associate its nerve; this is the simplicial set whose 0simplices are the objects, whose 1simplices are the morphisms, and whose nsimplices are the composable ntuples of morphisms. The face maps encode the domains and codomains of morphisms, the composition law, and the associativity property, while the degeneracies record information about the identities. This construction is the object part of a functor N: Cat1 → [ ∆ op,Set] from the category of categories and functors, to the category of simplicial sets. This functor is fully faithful and has a left adjoint. It arises in a natural way, as the “singular functor ” (see Section 1 below) of the inclusion
ON THE REGULAR REPRESENTATION OF AN (ESSENTIALLY) FINITE
, 907
"... Abstract. The regular representation of an essentially finite 2group G in the 2category 2Vectk of (Kapranov and Voevodsky) 2vector spaces is defined and cohomology invariants classifying it computed. It is next shown that all homcategories in Rep2Vectk G are 2vector spaces under quite standard ..."
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Abstract. The regular representation of an essentially finite 2group G in the 2category 2Vectk of (Kapranov and Voevodsky) 2vector spaces is defined and cohomology invariants classifying it computed. It is next shown that all homcategories in Rep2Vectk G are 2vector spaces under quite standard assumptions on the field k, and a formula giving the corresponding “intertwining numbers ” is obtained which proves they are symmetric. Finally, it is shown that the forgetful 2functor ω: Rep2Vectk G�2Vectk is representable with the regular representation as representing object. As a consequence we obtain a klinear equivalence between the 2vector space Vect G k of functors from the underlying groupoid of G to Vectk, on the one hand, and the klinear category End ω of pseudonatural endomorphisms of ω, on the other hand. We conclude that End ω is a 2vector space, and we (partially) describe a basis of it. 1.