Category Theory Map 1. Categories 1.1 Definition 1.2 Examples 1.2.1 Set 1.2.2 Grp 1.2.3 Ab 1.2.4 Top 1.2.5 Vect 1.2.6 Pos 1.2.7 Hask 1.3 Morphisms 1.3.1 Identity morphism 1.3.2 Composition of morphisms 1.3.3 Isomorphisms 1.3.4 Monomorphisms 1.3.5 Epimorphisms 1.3.6 Endomorphisms 1.3.7 Automorphisms 1.4 Initial and terminal objects 1.5 Zero objects 2. Functors 2.1 Definition 2.2 Examples 2.2.1 Identity functor 2.2.2 Constant functor 2.2.3 Forgetful functor 2.2.4 Free functor 2.2.5 Power set functor 2.2.6 Hom functor 2.3 Functor categories 2.4 Bifunctors 2.5 Contravariant functors 2.6 Representable functors 2.7 Adjoint functors 2.7.1 Left adjoint 2.7.2 Right adjoint 2.7.3 Adjunction 2.8 Equivalence of categories 2.9 Yoneda lemma 3. Natural Transformations 3.1 Definition 3.2 Examples 3.2.1 Identity natural transformation 3.2.2 Constant natural transformation 3.3 Horizontal composition 3.4 Vertical composition 3.5 Natural isomorphisms 4. Limits and Colimits 4.1 Limits 4.1.1 Definition 4.1.2 Examples 4.1.2.1 Products 4.1.2.2 Pullbacks 4.1.2.3 Equalizers 4.1.2.4 Inverse limits 4.2 Colimits 4.2.1 Definition 4.2.2 Examples 4.2.2.1 Coproducts 4.2.2.2 Pushouts 4.2.2.3 Coequalizers 4.2.2.4 Direct limits 4.3 Preservation of limits and colimits 5. Monoidal Categories 5.1 Definition 5.2 Examples 5.2.1 Cartesian monoidal categories 5.2.2 Cocartesian monoidal categories 5.2.3 Symmetric monoidal categories 5.2.4 Braided monoidal categories 5.3 Monoidal functors 5.4 Monoidal natural transformations 5.5 Coherence theorems 6. Enriched Categories 6.1 Definition 6.2 Examples 6.2.1 2-categories 6.2.2 Metric spaces 6.2.3 Simplicial sets 6.3 Enriched functors 6.4 Enriched natural transformations 7. Topos Theory 7.1 Definition 7.2 Examples 7.2.1 Set 7.2.2 Sheaves on a topological space 7.2.3 Etale topos 7.3 Subobject classifier 7.4 Power objects 7.5 Geometric morphisms 7.6 Classifying topoi 7.7 Grothendieck topoi 8. Abelian Categories 8.1 Definition 8.2 Examples 8.2.1 Ab 8.2.2 Mod_R (R-modules) 8.2.3 Sheaves of abelian groups 8.3 Kernels and cokernels 8.4 Images and coimages 8.5 Exact sequences 8.6 Projective and injective objects 8.7 Derived functors 8.7.1 Ext functors 8.7.2 Tor functors 8.8 Derived categories 8.9 Spectral sequences 9. Triangulated Categories 9.1 Definition 9.2 Examples 9.2.1 Derived categories 9.2.2 Stable homotopy categories 9.3 Triangles 9.4 Exact triangles 9.5 Octahedral axiom 9.6 Localization 10. Higher Category Theory 10.1 Strict n-categories 10.2 Weak n-categories 10.3 Infinity categories 10.4 Simplicial categories 10.5 Quasi-categories 10.6 Model categories 10.7 Homotopy hypothesis 11. Applications 11.1 Algebraic geometry 11.2 Algebraic topology 11.3 Homological algebra 11.4 Representation theory 11.5 Mathematical physics 11.6.1 Categorical quantum mechanics 11.6 Computer science 11.6.1 Type theory 11.6.2 Functional programming 11.6.3 Domain theory 11.7 Logic 11.7.1 Categorical logic 11.7.2 Topos theory as a foundation for mathematics 12. Monoidal Categories and Their Variants 12.1 Strict monoidal categories 12.2 Lax monoidal categories 12.3 Oplax monoidal categories 12.4 Braided monoidal categories 12.5 Symmetric monoidal categories 12.6 Ribbon categories 12.7 Tortile categories 12.8 Compact closed categories 12.9 Dagger categories 12.10 Frobenius algebras 12.11 Hopf algebras 12.12 Quantum groups 13. Fibrations and Opfibrations 13.1 Grothendieck fibrations 13.2 Grothendieck opfibrations 13.3 Cartesian morphisms 13.4 Opcartesian morphisms 13.5 Cleavages 13.6 Split fibrations 13.7 Split opfibrations 13.8 Fibrewise limits and colimits 13.9 Beck-Chevalley condition 13.10 Indexed categories 13.11 Slice categories 13.12 Comma categories 14. Profunctors and Distributors 14.1 Profunctors 14.2 Composition of profunctors 14.3 Distributors 14.4 Bimodules 14.5 Kan extensions 14.6 Weighted limits and colimits 14.7 Enriched profunctors 14.8 Monoidal profunctors 15. Sheaves and Stacks 15.1 Presheaves 15.2 Sheaves 15.3 Sites 15.4 Grothendieck topologies 15.5 Sheafification 15.6 Stacks 15.7 Gerbes 15.8 Descent theory 15.9 Grothendieck topoi 15.10 Classifying topoi 15.11 Topos-theoretic geometry 16. Operads and Multicategories 16.1 Operads 16.2 Symmetric operads 16.3 Cyclic operads 16.4 Modular operads 16.5 Multicategories 16.6 Symmetric multicategories 16.7 Generalized multicategories 16.8 Enriched operads 16.9 Homotopy theory of operads 16.10 Koszul duality for operads 17. 2-Categories and Bicategories 17.1 Strict 2-categories 17.2 Bicategories 17.3 Lax functors 17.4 Oplax functors 17.5 Pseudo functors 17.6 Natural transformations 17.7 Modifications 17.8 Adjunctions in 2-categories 17.9 Monads in 2-categories 17.10 Eilenberg-Moore construction 17.11 Kleisli construction 17.12 Monoidal bicategories 18. Double Categories and Equipments 18.1 Double categories 18.2 Vertical categories 18.3 Horizontal categories 18.4 Double functors 18.5 Double natural transformations 18.6 Equipments 18.7 Proarrow equipments 18.8 Framed bicategories 18.9 Double profunctors 19. Enriched Category Theory 19.1 V-categories 19.2 V-functors 19.3 V-natural transformations 19.4 V-limits and V-colimits 19.5 Weighted limits and colimits 19.6 V-monoidal categories 19.7 V-enriched adjunctions 19.8 V-enriched monads 19.9 V-enriched Kan extensions 19.10 V-enriched Yoneda lemma 19.11 V-enriched Day convolution 19.12 Change of base 20. Homotopical Algebra and Higher Structures 20.1 Model categories 20.2 Quillen adjunctions 20.3 Homotopy categories 20.4 Derived functors 20.5 Homotopy limits and colimits 20.6 Simplicial model categories 20.7 A_∞ categories 20.8 E_∞ categories 20.9 Differential graded categories 20.10 Stable infinity categories 20.11 Spectral categories 20.12 Factorization homology 21. Categorical Algebra 21.1 Monads 21.2 Algebras for a monad 21.3 Eilenberg-Moore category 21.4 Kleisli category 21.5 Comonads 21.6 Coalgebras for a comonad 21.7 Eilenberg-Moore category for comonads 21.8 Kleisli category for comonads 21.9 Bialgebras 21.10 Distributive laws 21.11 Lawvere theories 21.12 Monads in double categories 22. Categorical Logic and Type Theory 22.1 Cartesian closed categories 22.2 Lambda calculus 22.3 Simply typed lambda calculus 22.4 Dependent types 22.5 Intuitionistic type theory 22.6 Martin-Löf type theory 22.7 Homotopy type theory 22.8 Categorical semantics of type theories 22.9 Toposes as models of higher-order logic 22.10 Kripke-Joyal semantics 22.11 Sheaf semantics 22.12 Realizability toposes 23. Categorical Foundations 23.1 ETCS (Elementary Theory of the Category of Sets) 23.2 CCAF (Category of Categories as a Foundation) 23.3 Homotopy type theory as a foundation 23.4 Univalent foundations 23.5 Constructive set theory 23.6 Topos theory as a foundation 23.7 Categorical models of set theory 23.8 Grothendieck universes 23.9 Large categories 23.10 Accessible and locally presentable categories 23.11 Categorical logic as a foundation 23.12 Abstract Stone duality 24. Categorical Representation Theory 24.1 Tannakian categories 24.2 Rigid monoidal categories 24.3 Fusion categories 24.4 Modular tensor categories 24.5 Braided fusion categories 24.6 Pivotal categories 24.7 Spherical categories 24.8 Ribbon fusion categories 24.9 Module categories 24.10 Morita theory for tensor categories 24.11 Categorification of quantum groups 24.12 Categorical actions 25. Applied Category Theory 25.1 Categorical systems theory 25.2 Categorical control theory 25.3 Categorical databases 25.4 Categorical knowledge representation 25.5 Categorical linguistics 25.6 Categorical models of cognition 25.7 Categorical quantum mechanics 25.8 Categorical quantum information 25.9 Categorical network theory 25.10 Categorical game theory 25.11 Categorical economics 25.12 Categorical social choice theory