scholarly journals ASPECTS OF GLOBULAR HIGHER CATEGORY THEORY

2014 ◽  
Vol 90 (1) ◽  
pp. 172-173
Author(s):  
CAMELL KACHOUR
Keyword(s):  
Category Theory ◽  
Higher Category Theory

scholarly journals THH and Traces of Enriched Categories

2020 ◽  
Author(s):  
John D Berman
Keyword(s):  
Category Theory ◽  
Hochschild Homology ◽  
Independent Interest ◽  
Elementary Model ◽  
Enriched Category ◽  
Enriched Categories ◽  
Higher Category Theory ◽  
Model Independent ◽  
Universal Construction ◽  
Topological Hochschild Homology

Abstract We prove that topological Hochschild homology (THH) arises from a presheaf of circles on a certain combinatorial category, which gives a universal construction of THH for any enriched $\infty $-category. Our results rely crucially on an elementary, model-independent framework for enriched higher-category theory, which may be of independent interest. For those interested only in enriched category theory, read Sections 1.3 and 2.

scholarly journals Elementary remarks on units in monoidal categories

2008 ◽  
Vol 144 (1) ◽  
pp. 53-76 ◽  
Author(s):  
JOACHIM KOCK
Keyword(s):  
Category Theory ◽  
Compatibility Condition ◽  
Monoidal Category ◽  
Monoidal Categories ◽  
Alternative Definition ◽  
Monoidal Functor ◽  
Higher Category Theory ◽  
Definition Of

AbstractWe explore an alternative definition of unit in a monoidal category originally due to Saavedra: a Saavedra unit is a cancellable idempotent (in a 1-categorical sense). This notion is more economical than the usual notion in terms of left-right constraints, and is motivated by higher category theory. To start, we describe the semi-monoidal category of all possible unit structures on a given semi-monoidal category and observe that it is contractible (if non-empty). Then we prove that the two notions of units are equivalent in a strong functorial sense. Next, it is shown that the unit compatibility condition for a (strong) monoidal functor is precisely the condition for the functor to lift to the categories of units, and it is explained how the notion of Saavedra unit naturally leads to the equivalent non-algebraic notion of fair monoidal category, where the contractible multitude of units is considered as a whole instead of choosing one unit. To finish, the lax version of the unit comparison is considered. The paper is self-contained. All arguments are elementary, some of them of a certain beauty.

scholarly journals Hamiltonian Analysis for the Scalar Electrodynamics as 3BF Theory

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 620
Author(s):  
Tijana Radenković ◽  
Marko Vojinović
Keyword(s):  
Field Theory ◽  
Category Theory ◽  
Degrees Of Freedom ◽  
Canonical Quantization ◽  
Topological Field Theory ◽  
Dynamic Constraints ◽  
Specific Choice ◽  
Higher Category Theory ◽  
Topological Field ◽  
Hamiltonian Analysis

The higher category theory can be employed to generalize the B F action to the so-called 3 B F action, by passing from the notion of a gauge group to the notion of a gauge 3-group. The theory of scalar electrodynamics coupled to Einstein–Cartan gravity can be formulated as a constrained 3 B F theory for a specific choice of the gauge 3-group. The complete Hamiltonian analysis of the 3 B F action for the choice of a Lie 3-group corresponding to scalar electrodynamics is performed. This analysis is the first step towards a canonical quantization of a 3 B F theory, an important stepping stone for the quantization of the complete scalar electrodynamics coupled to Einstein–Cartan gravity formulated as a 3 B F action with suitable simplicity constraints. It is shown that the resulting dynamic constraints eliminate all propagating degrees of freedom, i.e., the 3 B F theory for this choice of a 3-group is a topological field theory, as expected.

TOWARDS NONCOMMUTATIVE COMPUTING

2000 ◽  
Vol 14 (22n23) ◽  
pp. 2451-2454
Author(s):  
G. F. MASCARI
Keyword(s):  
Information Processing ◽  
Quantum Information ◽  
Category Theory ◽  
Quantum Information Processing ◽  
Mathematical Structures ◽  
Structure And Dynamics ◽  
Physical Experiments ◽  
Higher Dimensional ◽  
Higher Category Theory ◽  
Categorical Constructions

This paper presents first steps of an approach to quantum information processing in the framework of higher category theory from a noncommutative mathematics perspective. The aim is to provide a unifying theory for the structure and dynamics of composite quantum information processing systems, such that states, evolution, entanglement, decoherence are modeled by abstract categorical constructions and vice versa new mathematical structures arising from higher dimensional algebra could be "tested" as computational schemes and possibly realized by physical experiments.

scholarly journals Cofree compositions of coalgebras

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Stefan Forcey ◽  
Aaron Lauve ◽  
Frank Sottile
Keyword(s):  
Hopf Algebra ◽  
Category Theory ◽  
The Other ◽  
Nous Concluons ◽  
Nous Montrons ◽  
Higher Category Theory ◽  
International Audience ◽  
Graded Coalgebra

International audience We develop the notion of the composition of two coalgebras, which arises naturally in higher category theory and the theory of species. We prove that the composition of two cofree coalgebras is cofree and give conditions which imply that the composition is a one-sided Hopf algebra. These conditions hold when one coalgebra is a graded Hopf operad $\mathcal{D}$ and the other is a connected graded coalgebra with coalgebra map to $\mathcal{D}$. We conclude with examples of these structures, where the factor coalgebras have bases indexed by the vertices of multiplihedra, composihedra, and hypercubes. Nous développons la notion de composition de coalgèbres, qui apparaît naturellement dans la théorie des catégories d'ordre supérieur et dans la théorie des espèces. Nous montrons que la composée de deux coalgèbres colibres est colibre et nous donnons des conditions qui impliquent que la composée est une algèbre de Hopf unilatérale. Ces conditions sont valables quand l'une des coalgèbres est une opérade de Hopf graduée $\mathcal{D}$ et l'autre est une coalgèbre graduée connexe avec un morphisme vers $\mathcal{D}$. Nous concluons avec des exemples de ces structures, où les coalgèbres composées ont des bases indexées par les sommets de multiplièdres, de composièdres, et d'hypercubes.

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