2-Dimensional Categories

Johnson, Niles; Yau, Donald

doi:10.48550/arxiv.2002.06055

Cited by 12 publications

(16 citation statements)

References 39 publications

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“…By Proposition 3.27, Proposition 3.30, it is also fully faithful on both 1-mrophisms and 2-morphisms. Then the Whitehead theorem for 2-categories (see Theorem 7.5.8 in [JY20]) implies that this 2-functor is a 2-equivalence.…”

Section: Canonical Constructionmentioning

confidence: 95%

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Enriched monoidal categories I: centers

Kong¹,

Yuan²,

Zhang³

et al. 2021

Preprint

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This work is the first one in a series, in which we develop a mathematical theory of enriched (braided) monoidal categories and their representations. In this work, we introduce the notion of the E 0 -center (E 1 -center or E 2 -center) of an enriched (monoidal or braided monoidal) category, and compute the centers explicitly when the enriched (braided monoidal or monoidal) categories are obtained from the canonical constructions. These centers have important applications in the mathematical theory of gapless boundaries of 2+1D topological orders and that of topological phase transitions in physics. They also play very important roles in the higher representation theory, which is the focus of the second work in the series.

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Section: Canonical Constructionmentioning

confidence: 95%

“…In this section, we recall some basic notions in 2-categories and examples, and set the notations along the way. We refer the reader to [JY20] for a detailed introduction to 2-categories and bicategories.…”

Section: -Categoriesmentioning

confidence: 99%

Enriched monoidal categories I: centers

Kong¹,

Yuan²,

Zhang³

et al. 2021

Preprint

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“…In this article, 2-category will always mean a weak 2-category/bicategory which is locally idempotent complete, and a C * /W * 2-category will always mean a weak C * /W * 2-category which is locally orthogonal projection complete. We refer the reader to [JY20] for background on 2-categories and to [CPJP21] for background on C * /W * 2-categories. We refer the reader to [HV19] or [CPJP21] for a detailed discussion of the graphical calculus of string diagrams for 2-categories.…”

Section: Preliminariesmentioning

confidence: 99%

Q-system completion is a 3-functor

Chen,

Penneys

2021

Preprint

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Q-systems are unitary versions of Frobenius algebra objects which appeared in the theory of subfactors. In recent joint work with R. Hernández Palomares and C. Jones, the authors defined a notion of Q-system completion for C * /W * 2-categories, which is a unitary version of a higher idempotent completion in the spirit of Douglas-Reutter and Gaiotto-Johnson-Freyd. In this article, we prove that Q-system completion is a † 3-functor on the † 3-category of C * /W * 2-categories. We also prove that Q-system completion satisfies a universal property analogous to the universal property satisfied by idempotent completion for 1-categories.

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“…Proof. Recall that in order for π R : N A K → Cat to be a 2-functor, it must be an assignment of 0-cells (objects), 1-cells, and 2-cells in N A K to those in Cat that strictly preserves identity 1-cells, identity 2-cells, vertical compositions of 2-cells, and horizontal compositions of 1-cells and of 2-cells [21].…”

Section: Bimodules and A Noncommutative Functor Of Pointsmentioning

confidence: 99%

Functor of Points and Height Functions for Noncommutative Arakelov Geometry

Lima,

Marcolli

2020

Preprint

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We propose a notion of functor of points for noncommutative spaces, valued in categories of bimodules, and endowed with an action functional determined by a notion of hermitian structures and height functions, modeled on an interpretation of the classical functor of points as a physical sigma model. We discuss different choices of such height functions, based on different notions of volumes and traces, including one based on the Hattori-Stallings rank. We show that the height function determines a dynamical time evolution on an algebra of observables associated to our functor of points. We focus in particular the case of noncommutative arithmetic curves, where the relevant algebras are sums of matrix algebras over division algebras over number fields, and we discuss a more general notion of noncommutative arithmetic spaces in higher dimensions, where our approach suggests an interpretation of the Jones index as a height function.

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2-Dimensional Categories

Cited by 12 publications

References 39 publications

Enriched monoidal categories I: centers

Enriched monoidal categories I: centers

Q-system completion is a 3-functor

Functor of Points and Height Functions for Noncommutative Arakelov Geometry

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