Transitive $2$-representations of finitary $2$-categories

Mazorchuk, Volodymyr; Miemietz, Vanessa

doi:10.1090/tran/6583

Cited by 58 publications

(239 citation statements)

References 25 publications

(34 reference statements)

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“…Here we also note that many results in [40]- [44] assume that a certain numerical condition is satisfied. This assumption was rendered superfluous by [45,Proposition 1].…”

Section: Theorem 11 Every Weakly Fiat 2-category With Strongly Regulamentioning

confidence: 82%

“…In this subsection we overview the weak Jordan-Hölder theory for additive 2-representations of finitary 2-categories developed in [44,Section 4]. Here simple transitive 2-representations play a crucial role.…”

Section: Weak Jordan-hölder Theorymentioning

confidence: 99%

“…Here simple transitive 2-representations play a crucial role. We start with the following observation which is just a variation of Lemma 2, see [44,Lemma 4].…”

Section: Weak Jordan-hölder Theorymentioning

confidence: 99%

“…When A does not have any non-zero projective-injective module, the approach of [44,47,48] After all the cases listed above, the following question is rather natural:…”

Section: Theorem 13 Let a Be A Basic Connected K-algebra Which Has A mentioning

confidence: 99%

“…Despite of the fact that the 2-category of 2-representations of an abstract finitary 2-category is much more complicated than the category of modules over a finite dimensional algebra (in particular, it is not abelian), it turned out that, in many cases, it is possible to construct, compare and even classify various classes of 2-representations. These kinds of problems were recently studied in a number of papers, see [32,[37][38][39][44][45][46][47][48]58]. The aim of the present article is to give an overview of these results with a particular emphasis on open problems and future challenges.…”

Section: Introductionmentioning

confidence: 99%

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Classification problems in 2-representation theory

Mazorchuk

2017

São Paulo J. Math. Sci.

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“…Here we also note that many results in [40]- [44] assume that a certain numerical condition is satisfied. This assumption was rendered superfluous by [45,Proposition 1].…”

Section: Theorem 11 Every Weakly Fiat 2-category With Strongly Regulamentioning

confidence: 82%

Section: Weak Jordan-hölder Theorymentioning

confidence: 99%

“…Here simple transitive 2-representations play a crucial role. We start with the following observation which is just a variation of Lemma 2, see [44,Lemma 4].…”

Section: Weak Jordan-hölder Theorymentioning

confidence: 99%

“…When A does not have any non-zero projective-injective module, the approach of [44,47,48] After all the cases listed above, the following question is rather natural:…”

Section: Theorem 13 Let a Be A Basic Connected K-algebra Which Has A mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Classification problems in 2-representation theory

Mazorchuk

2017

São Paulo J. Math. Sci.

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Group actions on 2-categories

2018

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We study actions of discrete groups on 2-categories. The motivating examples are actions on the 2-category of representations of finite tensor categories and their relation with the extension theory of tensor categories by groups. Associated to a group action on a 2-category, we construct the 2category of equivariant objects. We also introduce the G-equivariant notions of pseudofunctor, pseudonatural transformation and modification. Our first main result is a coherence theorem for 2-categories with an action of a group. For a 2-category B with an action of a group G, we construct a braided G-crossed monoidal category ZG(B) with trivial component the Drinfeld center of B. We prove that, in the case of a G-action on the 2-category of representation of a tensor category C, the 2-category of equivariant objects is biequivalent to the module categories over an associated G-extension of C. Finally, we prove that the center of the equivariant 2-category is monoidally equivalent to the equivariantization of a relative center, generalizing results obtained in [8].

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Fiat categorification of the symmetric inverse semigroup $$\textit{IS}_n$$ IS n and the semigroup $$F^*_n$$ F n ∗

Martin

Mazorchuk

2017

Semigroup Forum

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Starting from the symmetric group S n , we construct two fiat 2-categories. One of them can be viewed as the fiat "extension" of the natural 2-category associated with the symmetric inverse semigroup (considered as an ordered semigroup with respect to the natural order). This 2-category provides a fiat categorification for the integral semigroup algebra of the symmetric inverse semigroup. The other 2-category can be viewed as the fiat "extension" of the 2-category associated with the maximal factorizable subsemigroup of the dual symmetric inverse semigroup (again, considered as an ordered semigroup with respect to the natural order). This 2-category provides a fiat categorification for the integral semigroup algebra of the maximal factorizable subsemigroup of the dual symmetric inverse semigroup.

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Transitive $2$-representations of finitary $2$-categories

Cited by 58 publications

References 25 publications

Classification problems in 2-representation theory

Classification problems in 2-representation theory

Group actions on 2-categories

Fiat categorification of the symmetric inverse semigroup $$\textit{IS}_n$$ IS n and the semigroup $$F^*_n$$ F n ∗

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