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39
Fibrations of groupoids
 J. Algebra
, 1970
"... theory, and change of base for groupoids and multiple ..."
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Cited by 24 (15 self)
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theory, and change of base for groupoids and multiple
An introduction to ncategories
 In 7th Conference on Category Theory and Computer Science
, 1997
"... ..."
Pseudo algebras and pseudo double categories
 J. Homotopy Relat. Struct
"... Abstract. As an example of the categorical apparatus of pseudo algebras over 2theories, we show that pseudo algebras over the 2theory of categories can be viewed as pseudo double categories with folding or as appropriate 2functors into bicategories. Foldings are equivalent to connection pairs, an ..."
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Cited by 19 (2 self)
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Abstract. As an example of the categorical apparatus of pseudo algebras over 2theories, we show that pseudo algebras over the 2theory of categories can be viewed as pseudo double categories with folding or as appropriate 2functors into bicategories. Foldings are equivalent to connection pairs, and also to thin structures if the vertical and horizontal morphisms coincide. In a sense, the squares of a double category with folding are determined in a functorial way by the 2cells of the horizontal 2category. As a special case, strict 2algebras with one object and everything invertible are crossed modules under a group.
Crossed Complexes And Homotopy Groupoids As Non Commutative Tools For Higher Dimensional LocalToGlobal Problems
"... We outline the main features of the definitions and applications of crossed complexes and cubical #groupoids with connections. ..."
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Cited by 18 (7 self)
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We outline the main features of the definitions and applications of crossed complexes and cubical #groupoids with connections.
Precategories for Combining Probabilistic Automata
 Electronic Notes in Theoretical Computer Science
, 1999
"... A relaxed notion of category is presented having in mind the categorical caracterization of the mechanisms for combining probabilistic automata, since the composition of the appropriate morphisms is not always defined. A detailed discussion of the required notion of morphism is provided. The partial ..."
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Cited by 10 (6 self)
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A relaxed notion of category is presented having in mind the categorical caracterization of the mechanisms for combining probabilistic automata, since the composition of the appropriate morphisms is not always defined. A detailed discussion of the required notion of morphism is provided. The partiality of composition of such morphisms is illustrated at the abstract level of countable probability spaces. The relevant fragment of the theory of the proposed precategories is developed, including (constrained) products and Cartesian liftings. Precategories are precisely placed in the universe of neocategories. Some classical results from category theory are shown to carry over to precategories. Other results are shown not to hold in general. As an application, the precategorical universal constructs are used for characterizing the basic mechanisms for combining probabilistic automata: aggregation, interconnection and state constraining. Mathematics Subject Classifications: 18A10 68Q75. Ke...
Homotopy Theory, and Change of Base for Groupoids and Multiple Groupoids
, 1996
"... This survey article shows how the notion of "change of base", used in some applications to homotopy theory of the fundamental groupoid, has surprising higher dimensional analogues, through the use of certain higher homotopy groupoids with values in forms of multiple groupoids. ..."
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Cited by 5 (5 self)
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This survey article shows how the notion of "change of base", used in some applications to homotopy theory of the fundamental groupoid, has surprising higher dimensional analogues, through the use of certain higher homotopy groupoids with values in forms of multiple groupoids.
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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Cited by 5 (1 self)
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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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Cited by 4 (0 self)
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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.
An Invitation to Higher Gauge Theory
, 2010
"... In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2connections on 2bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge ‘2group’. We focus on 6 examples. First, every abelian Lie gr ..."
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Cited by 3 (2 self)
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In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2connections on 2bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge ‘2group’. We focus on 6 examples. First, every abelian Lie group gives a Lie 2group; the case of U(1) yields the theory of U(1) gerbes, which play an important role in string theory and multisymplectic geometry. Second, every group representation gives a Lie 2group; the representation of the Lorentz group on 4d Minkowski spacetime gives the Poincaré 2group, which leads to a spin foam model for Minkowski spacetime. Third, taking the adjoint representation of any Lie group on its own Lie algebra gives a ‘tangent 2group’, which serves as a gauge 2group in 4d BF theory, which has topological gravity as a special case. Fourth, every Lie group has an ‘inner automorphism 2group’, which serves as the gauge group in 4d BF theory with cosmological constant term. Fifth, every Lie group has an ‘automorphism 2group’, which plays an important role in the theory of nonabelian gerbes. And sixth, every compact simple Lie group gives a ‘string 2group’. We also touch upon higher structures such as the ‘gravity 3group’, and the Lie 3superalgebra that governs 11dimensional supergravity. 1