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Category theory

About: Category theory is a research topic. Over the lifetime, 2098 publications have been published within this topic receiving 52773 citations.


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Book
01 Jan 2009
TL;DR: In this paper, a general introduction to higher category theory using the formalism of "quasicategories" or "weak Kan complexes" is provided, and a few applications to classical topology are included.
Abstract: This purpose of this book is twofold: to provide a general introduction to higher category theory (using the formalism of "quasicategories" or "weak Kan complexes"), and to apply this theory to the study of higher versions of Grothendieck topoi. A few applications to classical topology are included.

1,879 citations

Book
01 Jan 1986
TL;DR: In this article, Cartesian closed categories and Calculus are used to represent Numerical functions in various categories and to describe the relation between categories. But they do not specify the topology of the categories.
Abstract: Preface Part I. Introduction to Category Theory: Part II. Cartesian Closed Categories and Calculus: Part III. Type Theory and Toposes: Part IV. Representing Numerical Functions in Various Categories Bibliography Author index Subject index.

1,388 citations

MonographDOI
26 Aug 1994
TL;DR: The Handbook of Categorical Algebra is intended to give, in three volumes, a rather detailed account of what, ideally, everybody working in category theory should know, whatever the specific topic of research they have chosen.
Abstract: The Handbook of Categorical Algebra is intended to give, in three volumes, a rather detailed account of what, ideally, everybody working in category theory should know, whatever the specific topic of research they have chosen. The book is planned also to serve as a reference book for both specialists in the field and all those using category theory as a tool. Volume 3 begins with the essential aspects of the theory of locales, proceeding to a study in chapter 2 of the sheaves on a locale and on a topological space, in their various equivalent presentations: functors, etale maps or W-sets. Next, this situation is generalized to the case of sheaves on a site and the corresponding notion of Grothendieck topos is introduced. Chapter 4 relates the theory of Grothendieck toposes with that of accessible categories and sketches, by proving the existence of a classifying topos for all coherent theories.

1,321 citations

Posted Content
TL;DR: In this article, the authors define new invariants of 3d Calabi-Yau categories endowed with a stability structure, which are elements of quantum tori over a version of the Grothendieck ring of varieties over the ground field.
Abstract: We define new invariants of 3d Calabi-Yau categories endowed with a stability structure. Intuitively, they count the number of semistable objects with fixed class in the K-theory of the category ('number of BPS states with given charge' in physics language). Formally, our motivic DT-invariants are elements of quantum tori over a version of the Grothendieck ring of varieties over the ground field. Via the quasi-classical limit 'as the motive of affine line approaches to 1' we obtain numerical DT-invariants which are closely related to those introduced by Behrend. We study some properties of both motivic and numerical DT-invariants including the wall-crossing formulas and integrality. We discuss the relationship with the mathematical works (in the non-triangulated case) of Joyce, Bridgeland and Toledano-Laredo, as well as with works of physicists on Seiberg-Witten model (string junctions), classification of N=2 supersymmetric theories (Cecotti-Vafa) and structure of the moduli space of vector multiplets. Relating the theory of 3d Calabi-Yau categories with distinguished set of generators (called cluster collection) with the theory of quivers with potential we found the connection with cluster transformations and cluster varieties (both classical and quantum).

1,087 citations

Book
15 Jan 1996
TL;DR: In this article, the authors introduce homotopic algebra and define the notion of simplicial sets, derived categories and derived functors, and triangulated categories for homotopy algebra.
Abstract: I. Simplicial Sets.- II. Main Notions of the Category Theory.- III. Derived Categories and Derived Functors.- IV. Triangulated Categories.- V. Introduction to Homotopic Algebra.- References.

1,043 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202329
202264
2021105
2020116
2019113
2018102