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Oriented Category Theory
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
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
Category Theory Lecture Notes by McGill University
This note covers the following topics, Preliminaries, Categories, Properties of objects and arrows, Functors, Diagrams and naturality, Products and sums, Cartesian closed categories, Limits and colimits, Adjoints, Triples, Toposes and Categories with monoidal structure.
Author(s): Department of Mathematics and Statistics,McGill University
133 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 Programming Language Oriented Introduction
This book explains the following topics, Categories, functors, natural transformations, String diagrams, Kan extensions, Algebras, coalgebras, bialgebras, Lambdacalculus and categories.
Author(s): Pierre-Louis Curien
145 Pages
Category Theory by Prof Dr B Pareigis
This book explains the following topics related to Category Theory, Foundations, Graphs, Monoids, Categories, Constructions on categories, Functors, Special types of functors, Natural transformations, Representable functors and the Yoneda Lemma, Terminal and initial objects, The extension principle, Isomorphisms, Monomorphisms and epimorphisms, Products, Adjoint functors and monads.
Author(s): Prof. Dr. B. Pareigis
90 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