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Category Theory

New submissions

[ total of 6 entries: 1-6 ]
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New submissions for Fri, 12 Apr 24

[1]  arXiv:2404.07273 [pdf, ps, other]
Title: Combinatorics of higher-categorical diagrams
Comments: 335 pages
Subjects: Category Theory (math.CT); Algebraic Topology (math.AT); Combinatorics (math.CO)

This is a book on higher-categorical diagrams, including pasting diagrams. It aims to provide a thorough and modern reference on the subject, collecting, revisiting and expanding results scattered across the literature, informed by recent advances and practical experience with higher-dimensional diagram rewriting.
We approach the subject as a kind of directed combinatorial topology: a diagram is a map from a "directed cell complex", encoded combinatorially as a face poset together with orientation data. Unlike previous expositions, we adopt from the beginning a functorial viewpoint, focussing on morphisms and categorical constructions. We do not tie ourselves to a specific model of higher categories, and instead treat diagrams as independent combinatorial structures that admit functorial interpretations in various contexts.
Topics covered include the theory of layerings of diagrams; acyclicity properties and their consequences; constructions including Gray products, suspensions, and joins; special shapes such as globes, oriented simplices, cubes, and positive opetopes; the interpretation of diagrams in strict omega-categories and their geometric realisation as simplicial and CW complexes; and Steiner's theory of directed chain complexes.

Cross-lists for Fri, 12 Apr 24

[2]  arXiv:2404.07583 (cross-list from math.AG) [pdf, ps, other]
Title: Length of triangulated categories
Comments: 24 pages, comments are welcome!
Subjects: Algebraic Geometry (math.AG); Category Theory (math.CT); Rings and Algebras (math.RA); Representation Theory (math.RT)

We introduce the notion of length of triangulated categories, and we compute the length of the derived category ${\rm D}^{\rm b}(C)$ of a smooth projective curve $C$ and classify all finite length thick subcategories of ${\rm D}^{\rm b}(C)$. We also discuss the length of (1) derived categories of finite dimensional representations of ADE quivers, (2) derived categories of some singular varieties and (3) Krah's phantom subcategories.

[3]  arXiv:2404.07854 (cross-list from cs.LO) [pdf, other]
Title: Reflexive graph lenses in univalent foundations
Comments: 52 pages
Subjects: Logic in Computer Science (cs.LO); Category Theory (math.CT); Logic (math.LO)

Martin-L\"of's identity types provide a generic (albeit opaque) notion of identification or "equality" between any two elements of the same type, embodied in a canonical reflexive graph structure $(=_A, \mathbf{refl})$ on any type $A$. The miracle of Voevodsky's univalence principle is that it ensures, for essentially any naturally occurring structure in mathematics, that this the resultant notion of identification is equivalent to the type of isomorphisms in the category of such structures. Characterisations of this kind are not automatic and must be established one-by-one; to this end, several authors have employed reflexive graphs and displayed reflexive graphs to organise the characterisation of identity types.
We contribute reflexive graph lenses, a new family of intermediate abstractions lying between families of reflexive graphs and displayed reflexive graphs that simplifies the characterisation of identity types for complex structures. Every reflexive graph lens gives rise to a (more complicated) displayed reflexive graph, and our experience suggests that many naturally occurring displayed reflexive graphs arise in this way. Evidence for the utility of reflexive graph lenses is given by means of several case studies, including the theory of reflexive graphs itself as well as that of polynomial type operators. Finally, we exhibit an equivalence between the type of reflexive graph fibrations and the type of univalent reflexive graph lenses.

Replacements for Fri, 12 Apr 24

[4]  arXiv:2205.02142 (replaced) [pdf, ps, other]
Title: The Sup Connective in IMALL: A Categorical Semantics
Subjects: Logic in Computer Science (cs.LO); Category Theory (math.CT); Logic (math.LO)
[5]  arXiv:2309.01468 (replaced) [pdf, ps, other]
Title: Surprising occurrences of order structures in mathematics
Authors: Gunnar Fløystad
Comments: 26 pages, minor corrections and improvements
Subjects: History and Overview (math.HO); Commutative Algebra (math.AC); Combinatorics (math.CO); Category Theory (math.CT); Rings and Algebras (math.RA)
[6]  arXiv:2403.10479 (replaced) [pdf, other]
Title: Complete equational theories for classical and quantum Gaussian relations
Comments: small fixes
Subjects: Logic in Computer Science (cs.LO); Category Theory (math.CT); Quantum Physics (quant-ph)
[ total of 6 entries: 1-6 ]
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