Braided Tensor Categories

Joyal, André; Street, Ross

doi:10.1006/aima.1993.1055

Cited by 853 publications

(936 citation statements)

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“…Braided tensor categories. The notion of braided tensor category is due to Joyal and Street [21]. Assume that both C and D are braided tensor categories.…”

Section: 4mentioning

confidence: 99%

The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

Lehrer

Zhang

2016

Commun. Math. Phys.

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Abstract. Let U q (g) be the quantum supergroup of gl m|n or the modified quantum supergroup of osp m|2n over the field of rational functions in q, and let V q be the natural module for U q (g). There exists a unique tensor functor, associated with V q , from the category of ribbon graphs to the category of finite dimensional representations of U q (g), which preserves ribbon category structures. We show that this functor is full in the cases g = gl m|n or osp 2ℓ+1|2n . For g = osp 2ℓ|2n , we show that the space Hom Uq(g) (V ⊗r q , V ⊗s q ) is spanned by images of ribbon graphs if r + s < 2ℓ(2n + 1). The proofs involve an equivalence of module categories for two versions of the quantisation of U(g).

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“…Braided tensor categories. The notion of braided tensor category is due to Joyal and Street [21]. Assume that both C and D are braided tensor categories.…”

Section: 4mentioning

confidence: 99%

The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

Lehrer

Zhang

2016

Commun. Math. Phys.

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“…More structural approaches to coherence may be found in [24,27,35]-these motivate the approach taken in Sect. 7 onwards.…”

Section: Coherence For Associativity-the Unitless Settingmentioning

confidence: 90%

“…In particular, Joyal and Street phrased MacLane's theorem as an equivalence between the free monoidal category on a category and the free strict monoidal category on a category (see [24] for details and extensions of this approach).…”

Section: The Following Diagram Commutesmentioning

confidence: 99%

Coherence and strictification for self-similarity

Hines

2016

J. Homotopy Relat. Struct.

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This paper studies questions of coherence and strictification related to selfsimilarity-the identity S ∼ = S ⊗ S in a semi-monoidal category. Based on Saavedra's theory of units, we first demonstrate that strict self-similarity cannot simultaneously occur with strict associativity-i.e. no monoid may have a strictly associative (semi-) monoidal tensor, although many monoids have a semi-monoidal tensor associative up to isomorphism. We then give a simple coherence result for the arrows exhibiting self-similarity and use this to describe a 'strictification procedure' that gives a semimonoidal equivalence of categories relating strict and non-strict self-similarity, and hence monoid analogues of many categorical properties. Using this, we characterise a class of diagrams (built from the canonical isomorphisms for the relevant tensors, together with the isomorphisms exhibiting the self-similarity) that are guaranteed to commute, and give a simple intuitive interpretation of this characterisation.

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“…is the following composition: In a similar way it is possible to introduce a graded version Z (H) of the centre Z(H) of a categorical group (see [9,Example 2.3]) and it is easy to observe that π 1 …”

Section: Preliminariesmentioning

confidence: 99%

Equivariant Extensions of Categorical Groups

Garzón

Río

2005

Appl Categor Struct

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Abstract. If is a group, then the category of -graded categorical groups is equivalent to the category of categorical groups supplied with a coherent left-action from . In this paper we use this equivalence and the homotopy classification of graded categorical groups and their homomorphisms to develop a theory of extensions of categorical groups when a fixed group of operators is acting. For this kind of extensions we show a suitable Schreier's theory and a precise theorem of classification, including obstruction theory, which generalizes both known results when the group of operators is trivial (categorical group extensions theory) or when the involved categorical groups are discrete (equivariant group extensions theory).

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Braided Tensor Categories

Cited by 853 publications

References 0 publications

The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

Coherence and strictification for self-similarity

Equivariant Extensions of Categorical Groups

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