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Higher dimensional algebra V: 2groups
 Theory Appl. Categ
"... A 2group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to tw ..."
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Cited by 25 (2 self)
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A 2group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to two, which we call ‘weak ’ and ‘coherent ’ 2groups. A weak 2group is a weak monoidal category in which every morphism has an inverse and every object x has a ‘weak inverse’: an object y such that x ⊗ y ∼ = 1 ∼ = y ⊗ x. A coherent 2group is a weak 2group in which every object x is equipped with a specified weak inverse ¯x and isomorphisms ix: 1 → x ⊗ ¯x, ex: ¯x ⊗ x → 1 forming an adjunction. We describe 2categories of weak and coherent 2groups and an ‘improvement ’ 2functor that turns weak 2groups into coherent ones, and prove that this 2functor is a 2equivalence of 2categories. We internalize the concept of coherent 2group, which gives a quick way to define Lie 2groups. We give a tour of examples, including the ‘fundamental 2group ’ of a space and various Lie 2groups. We also explain how coherent 2groups can be classified in terms of 3rd cohomology classes in group cohomology. Finally, using this classification, we construct for any connected and simplyconnected compact simple Lie group G a family of 2groups G � ( � ∈ Z) having G as its group of objects and U(1) as the group of automorphisms of its identity object. These 2groups are built using Chern–Simons theory, and are closely related to the Lie 2algebras g � ( � ∈ R) described in a companion paper. 1 1
Homotopytheoretic aspects of 2–monads
 J. Homotopy Relat. Struct
"... We study 2monads and their algebras using a Catenriched version of Quillen model categories, emphasizing the parallels between the homotopical and 2categorical points of view. Every 2category with finite limits and colimits has a canonical model structure in which the weak equivalences are the e ..."
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Cited by 6 (0 self)
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We study 2monads and their algebras using a Catenriched version of Quillen model categories, emphasizing the parallels between the homotopical and 2categorical points of view. Every 2category with finite limits and colimits has a canonical model structure in which the weak equivalences are the equivalences; we use these to construct more interesting model structures on 2categories, including a model structure on the 2category of algebras for a 2monad T, and a model structure on a 2category of 2monads on a fixed 2category K. 1
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
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.
A 2categories companion
"... Abstract. This paper is a rather informal guide to some of the basic theory of 2categories and bicategories, including notions of limit and colimit, 2dimensional universal algebra, formal category theory, and nerves of bicategories. 1. Overview and basic examples This paper is a rather informal gu ..."
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Abstract. This paper is a rather informal guide to some of the basic theory of 2categories and bicategories, including notions of limit and colimit, 2dimensional universal algebra, formal category theory, and nerves of bicategories. 1. Overview and basic examples This paper is a rather informal guide to some of the basic theory of 2categories and bicategories, including notions of limit and colimit, 2dimensional universal algebra, formal category theory, and nerves of bicategories. As is the way of these things, the choice of topics is somewhat personal. No attempt is made at either rigour or completeness. Nor is it completely introductory: you will not find a definition of bicategory; but then nor will you really need one to read it. In keeping with the philosophy of category theory, the morphisms between bicategories play more of a role than the bicategories themselves. 1.1. The key players. There are bicategories, 2categories, and Catcategories. The latter two are exactly the same (except that strictly speaking a Catcategory should have small homcategories, but that need not concern us here). The first two are nominally different — the 2categories are the strict bicategories, and not every bicategory is strict — but every bicategory is biequivalent to a strict one, and biequivalence is the right general notion of equivalence for bicategories and for 2categories. Nonetheless, the theories of bicategories, 2categories, and Catcategories have rather different flavours.
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 ..."
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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. 1
On The Monadicity Of Categories With Chosen Colimits
 THEORY APPL. CATEG
, 2000
"... There is a 2category JColim of small categories equipped with a choice of colimit for each diagram whose domain J lies in a given small class J of small categories, functors strictly preserving such colimits, and natural transformations. The evident forgetful 2functor from JColim to the 2ca ..."
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There is a 2category JColim of small categories equipped with a choice of colimit for each diagram whose domain J lies in a given small class J of small categories, functors strictly preserving such colimits, and natural transformations. The evident forgetful 2functor from JColim to the 2category Cat of small categories is known to be monadic. We extend this result by considering not just conical colimits, but general weighted colimits; not just ordinary categories but enriched ones; and not just small classes of colimits but large ones; in this last case we are forced to move from the 2category VCat of small Vcategories to Vcategories with objectset in some larger universe. In each case, the functors preserving the colimits in the usual "uptoisomorphism" sense are recovered as the pseudomorphisms between algebras for the 2monad in question.
FRAMED BICATEGORIES AND MONOIDAL FIBRATIONS MICHAEL SHULMAN
, 706
"... 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 as bimodules, spans, distributors, or parametrized spectra. Many bicategorical notions do not work well in these cases, becau ..."
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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 as bimodules, spans, distributors, or parametrized spectra. Many bicategorical notions do not work well in these cases, because the ‘morphisms between 0cells’, such as ring homomorphisms, are missing. We can include them by using a pseudo double category, but usually these morphisms also induce base change functors acting on the 1cells. We avoid complicated coherence problems by describing base change ‘nonalgebraically’, using categorical fibrations. The resulting ‘framed bicategories ’ assemble into 2categories, with attendant notions of equivalence, adjunction, and so on which are more appropriate for our examples than are the usual bicategorical ones. We then describe two ways to construct framed bicategories. One is an analogue of rings and bimodules which starts from one framed bicategory and builds another. The other starts from a ‘monoidal fibration’, meaning a parametrized family of monoidal categories, and produces an analogue of the framed bicategory of spans. Combining the two, we obtain a construction which includes both enriched and internal categories as special cases.
FRAMED BICATEGORIES AND MONOIDAL FIBRATIONS MICHAEL SHULMAN
, 706
"... 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 as bimodules, spans, distributors, or parametrized spectra. Many bicategorical notions do not work well in these cases, becau ..."
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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 as bimodules, spans, distributors, or parametrized spectra. Many bicategorical notions do not work well in these cases, because the ‘morphisms between 0cells’, such as ring homomorphisms, are missing. We can include them by using a pseudo double category, but usually these morphisms also induce base change functors acting on the 1cells. We avoid complicated coherence problems by describing base change ‘nonalgebraically’, using categorical fibrations. The resulting ‘framed bicategories ’ assemble into 2categories, with attendant notions of equivalence, adjunction, and so on which are more appropriate for our examples than are the usual bicategorical ones. We then describe two ways to construct framed bicategories. One is an analogue of rings and bimodules which starts from one framed bicategory and builds another. The other starts from a ‘monoidal fibration’, meaning a parametrized family of monoidal categories, and produces an analogue of