The combinatorics of n-categorical pasting

Johnson, Michael

doi:10.1016/0022-4049(89)90136-9

Cited by 66 publications

(41 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this situation an n-stage diagram in C is just a special sort of n-pasting diagram in C [16,24]. For example, a diagram over U 2 is the following pasting diagram: the diagrams over M 2 ) as an exercise.…”

Section: Coherence For Monoidal Globular Categoriesmentioning

confidence: 99%

Monoidal Globular Categories As a Natural Environment for the Theory of Weakn-Categories

Batanin¹

1998

Advances in Mathematics

188

450

View full text Add to dashboard Cite

We present a definition of weak |-categories based on a higher-order generalization of apparatus of operads.1998 Academic Press Contents.1. From non-symmetric to higher order operads. Monoidal globular categories.3. Some examples. 4.Coherence for monoidal globular categories. Coproducts in monoidal globular categories.6. Collections in globular categories. 7.Higher-order operads and their algebras.8. Contractibility and weak categories. Higher fundamental groupoids.10. Appendix: 2-operads and bicategories.

show abstract

Section: Coherence For Monoidal Globular Categoriesmentioning

confidence: 99%

Monoidal Globular Categories As a Natural Environment for the Theory of Weakn-Categories

Batanin¹

1998

Advances in Mathematics

188

450

View full text Add to dashboard Cite

show abstract

“…Luckily they are, thanks to associativity and the interchange law. This is a nontrivial result, called the 'pasting theorem' [60,77]. By this method, we can reduce the task of computing hol( ) for a large surface to the task of computing it for lots of small squares.…”

Section: -Connectionsmentioning

confidence: 99%

An invitation to higher gauge theory

2010

View full text Add to dashboard Cite

In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2-connections on 2-bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge '2-group'. We focus on 6 examples. First, every abelian Lie group gives a Lie 2-group; 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 2-group; the representation of the Lorentz group on 4d Minkowski spacetime gives the Poincaré 2-group, 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 2-group', which serves as a gauge 2-group in 4d BF theory, which has topological gravity as a special case. Fourth, every Lie group has an 'inner automorphism 2-group', which serves as the gauge group in 4d BF theory with cosmological constant term. Fifth, every Lie group has an 'automorphism 2-group', which plays an important role in the theory of nonabelian gerbes. And sixth, every compact simple Lie group gives a 'string 2-group'. We also touch upon higher structures such as the 'gravity 3-group', and the Lie 3-superalgebra that governs 11-dimensional supergravity.

show abstract

“…We essentially recover the examples constructed in earlier treatments such as [6], [9], [10] and [12]. Definition 3.1.…”

Section: Bases For Augmented Directed Complexesmentioning

confidence: 99%

“…In the past, ω-categories have been constructed from combinatorial structures such as pasting schemes, parity complexes or directed complexes; see Johnson [6], Power [9], Steiner [10] and Street [12]. The constructions are not really functorial, although there are ways to produce non-obvious morphisms; see Crans-Steiner [4].…”

Section: Introductionmentioning

confidence: 99%

Omega-categories and chain complexes

Steiner

2004

Homology, Homotopy and Applications

View full text Add to dashboard Cite

There are several ways to construct omega-categories from combinatorial objects such as pasting schemes or parity complexes. We make these constructions into a functor on a category of chain complexes with additional structure, which we call augmented directed complexes. This functor from augmented directed complexes to omega-categories has a left adjoint, and the adjunction restricts to an equivalence on a category of augmented directed complexes with good bases. The omega-categories equivalent to augmented directed complexes with good bases include the omega-categories associated to globes, simplexes and cubes; thus the morphisms between these omega-categories are determined by morphisms between chain complexes. It follows that the entire theory of omega-categories can be expressed in terms of chain complexes; in particular we describe the biclosed monoidal structure on omega-categories and calculate some internal homomorphism objects.

show abstract

The combinatorics of n-categorical pasting

Cited by 66 publications

References 4 publications

Monoidal Globular Categories As a Natural Environment for the Theory of Weakn-Categories

Monoidal Globular Categories As a Natural Environment for the Theory of Weakn-Categories

An invitation to higher gauge theory

Omega-categories and chain complexes

Contact Info

Product

Resources

About