Abstract Algebra Algebra Algebraic Geometry Algebraic Topology Applied Mathematics Arithmetic Geometry Basic Algebra Basic Mathematics Calculus Category Theory Classical Analysis Combinatorics Commutative Algebra Complex Algebra Complex Analysis Differential Algebra Differential Analysis Differential Calculus Differential Equations Differential Geometry Differential Topology Discrete Mathematics Elliptic Curves Fourier Analysis Functional Analysis Fractals Fractional Calculus Geometric Algebra Geometry Geometric Topology Groups Theory Graph Theory Harmonic Analysis Higher Algebra History of Mathematics Homological Algebra Integral Calculus K-theory Lie Algebra Linear Algebra Mathematical Analysis Mathematical Series Modern Geometry Multivairable Calculus Number Theory Numerical Analysis Probability Theory Real Analysis Rings and Fileds Riemannian Geometry Theorems in Calculus Topology Trigonometry Set Theory Constants & Numerical Sequences
Tensor Categorie (PDF 93P)
This note covers the following topics: Monoidal categories, The pentagon axiom, Basic properties of unit objects in monoidal categories, monoidal categories, Monoidal functors, equivalence of monoidal categories, Morphisms of monoidal functors, MacLane's strictness theorem, The MacLane coherence theorem, Invertible objects, Exactness of the tensor product, Semisimplicity of the unit object, Groupoids, Finite abelian categories and exact faithful functors, Fiber functors, Hopf algebras, Pointed tensor categories and pointed Hopf algebras, Chevalley's theorem, The Andruskiewitsch-Schneider conjecture, The Cartier-Kostant theorem, Pivotal categories and dimensions, Spherical categories and Grothendieck rings of semisimple tensor categories.
Author(s): P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik
393 Pages 
This book contains below topics, Note to the reader, Categories functors and natural transformations, Adjoints, Interlude on sets, Representables, Limits, Adjoints representables and limits, Index of notation.
Author(s): Tom Leinster
191 Pages
As categorical dimension increases, constructions such as cylinders, cones, and suspensions are no longer functorial in the old sense. These notes extend higher categories to oriented and antioriented categories. Compositions there commute only up to a coherent interchange law.
Author(s): David Gepner, Hadrian Heine
67 Pages
Parametrized higher category theory
This book contains below topics, Cocartesian fibrations and model categories of marked simplicial sets, Functor categories, Join and slice, Limits and colimits, Assembling Sslice categories from ordinary slice categories, Types of Sfibrations, Relative adjunctions, Parametrized colimits, Kan extensions, Yoneda lemma, Bousfield Kan formula, fibrant replacement.
Author(s): Jay Shah
83 Pages
This paper looks at diagrammatic category and keeps the language simple. It asks what diagrammatic is, how category fits, and what a student should notice. Count, size, and order in diagrammatic are compared with everyday numbers.
Author(s): Kenji Nakahira
99 Pages
An introduction to Category Theory by Valdis Laan
This note explains categories, Properties of morphisms and objects, Functors, Natural transformations, Limits and colimits, Adjunctions.
Author(s): Valdis Laan
52 Pages
Categorical homotopy theory by Emily Riehl
This PDF book covers the following topics related to Category Theory, All concepts are Kan extensions, Derived functors via deformations, Basic concepts of enriched category theory, The unreasonably effective bar construction, Homotopy limits and colimits, the practice, Weighted limits and colimits, Categorical tools for homotopy limit computations, Weighted homotopy limits and colimits, Derived enrichment, Weak factorization systems in model categories, Algebraic perspectives on the small object argument, Enriched factorizations and enriched lifting properties, A brief tour of Reedy category theory,. Preliminaries on quasicategories, Simplicial categories and homotopy coherence, Isomorphisms in quasicategories, A sampling of 2categorical aspects of quasicategory theory.
Author(s): Emily Riehl
292 Pages
Category theory, a branch of abstract algebra, has found many applications in mathematics, logic, and computer science. Like such fields as elementary logic and set theory, category theory provides a basic conceptual apparatus and a collection of formal methods useful for addressing certain kinds of commonly occurring formal and informal problems, particularly those involving structural and functional considerations. This course and nbsp, note is intended to acquaint students with these methods, and also to encourage them to reflect on the interrelations between category theory and the other basic formal disciplines.
Author(s): Steve Awodey
NA Pages
This note covers the following topics, and nbsp, Monoidal categories, The pentagon axiom, Basic properties of unit objects in monoidal categories, monoidal categories, Monoidal functors, equivalence of monoidal categories, Morphisms of monoidal functors, MacLanes strictness theorem, The MacLane coherence theorem, Invertible objects, Exactness of the tensor product, Semisimplicity of the unit object, Groupoids, Finite abelian categories and exact faithful functors, Fiber functors, Hopf algebras, Pointed tensor categories and pointed Hopf algebras, Chevalleys theorem, The AndruskiewitschSchneider conjecture, The CartierKostant theorem, Pivotal categories and dimensions, Spherical categories and Grothendieck rings of semisimple tensor categories.
Author(s): P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik
393 Pages