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{{Short description|1=Overview of and topical guide to category theory}}
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The following [[Outline (list)|outline]] is provided as an overview of and guide to [[category theory]], the area of study in [[mathematics]] that examines in an [[abstraction|abstract]] way the properties of particular mathematical concepts, by formalising them as collections of ''objects'' and ''arrows'' (also called [[morphism]]s, although this term also has a specific, non category-theoretical sense), where these collections satisfy certain basic conditions.  Many significant areas of mathematics can be formalised as categories, and the use of category theory allows many intricate and subtle mathematical results in these fields to be stated, and proved, in a much simpler way than without the use of categories.

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== Essence of category theory ==

* [[Category (mathematics)|Category]] 
* [[Functor]] 
* [[Natural transformation]]

== Branches of category theory ==
* [[Homological algebra]] 
* [[Diagram chasing]] 
* [[Topos|Topos theory]] 
* [[Enriched category]] theory 
* [[Higher category theory]] 
* [[Categorical logic]]
* [[Applied category theory]]

== Specific categories ==
*[[Category of sets]] 
**[[Concrete category]] 
*[[Category of vector spaces]] 
**[[Category of graded vector spaces]] 
*[[Category of chain complexes]] 
*[[Category of finite dimensional Hilbert spaces]] 
*[[Category of sets and relations]] 
*[[Category of topological spaces]] 
*[[Category of metric spaces]] 
*[[Category of preordered sets]] 
*[[Category of groups]] 
*[[Category of abelian groups]] 
*[[Category of rings]] 
*[[Category of magmas]] 

==Objects==
*[[Initial object]] 
*[[Terminal object]] 
*[[Zero object]] 
*[[Subobject]] 
*[[Group object]] 
*[[Magma object]] 
*[[Natural number object]] 
*[[Exponential object]]

==Morphisms==
{{Main|Morphism}}
 
*[[Epimorphism]] 
*[[Monomorphism]] 
*[[Zero morphism]] 
*[[Normal morphism]] 
*[[Dual (category theory)]] 
*[[Groupoid]] 
*[[Image (category theory)]] 
*[[Coimage]] 
*[[Commutative diagram]] 
*[[Cartesian morphism]] 
*[[Slice category]]

==Functors==
{{Main|Functor}}

*[[Isomorphism of categories]] 
*[[Natural transformation]] 
*[[Equivalence of categories]] 
*[[Subcategory]] 
*[[Faithful functor]] 
*[[Full functor]] 
*[[Forgetful functor]] 
*[[Yoneda lemma]] 
*[[Representable functor]] 
*[[Functor category]] 
*[[Adjoint functors]] 
**[[Galois connection]] 
**[[Pontryagin duality]] 
**[[Affine scheme]] 
*[[Monad (category theory)]] 
*[[Comonad]] 
*[[Combinatorial species]] 
*[[Exact functor]] 
*[[Derived functor]] 
*[[Dominant functor]] 
*[[Enriched functor]] 
*[[Kan extension|Kan extension of a functor]] 
*[[Hom functor]]

==Limits==
{{Main|Limit (category theory)}}

*[[Product (category theory)]] 
*[[Equaliser (mathematics)]] 
*[[Kernel (category theory)]] 
*[[Pullback (category theory)]]/[[fiber product]] 
*[[Inverse limit]] 
**[[Pro-finite group]] 
*[[Colimit]] 
**[[Coproduct]] 
**[[Coequalizer]] 
**[[Cokernel]] 
**[[Pushout (category theory)]] 
**[[Direct limit]] 
*[[Biproduct]] 
**[[Direct sum]]

==Additive structure==

*[[Preadditive category]] 
*[[Additive category]] 
*[[Pre-Abelian category]] 
*[[Abelian category]] 
**[[Exact sequence]] 
**[[Exact functor]] 
**[[Snake lemma]] 
***[[Nine lemma]] 
**[[Five lemma]] 
***[[Short five lemma]] 
**[[Mitchell's embedding theorem]] 
*[[Injective cogenerator]] 
*[[Derived category]] 
*[[Triangulated category]] 
*[[Model category]] 
*[[2-category]]

==Dagger categories==
{{Main|Dagger category}}

*[[Dagger symmetric monoidal category]] 
*[[Dagger compact category]] 
*[[Strongly ribbon category]]

==Monoidal categories==
{{Main|Monoidal category}}

*[[Closed monoidal category]] 
*[[Braided monoidal category]]

==Cartesian closed category==
*[[Topos]]
*[[Category of small categories]]

==Structure==
{{Main|Structure (category theory)}}

*[[Semigroupoid]] 
*[[Comma category]] 
*[[Localization of a category]] 
*[[Enriched category]] 
*[[Bicategory]]

==Topoi, toposes==

* [[Sheaf (mathematics)|Sheaf]] 
* [[Gluing axiom]]  
* [[Descent (category theory)]] 
* [[Grothendieck topology]] 
* [[Introduction to topos theory]] 
* [[Subobject classifier]] 
* [[Pointless topology]] 
* [[Heyting algebra]]

== History of category theory ==

*[[Category theory#Historical notes|History of category theory]]

== Persons influential in the field of category theory ==

=== Category theory scholars ===

*[[Saunders Mac Lane]]
*[[Samuel Eilenberg]]
*[[Max Kelly]]
*[[William Lawvere]]
*[[André Joyal]]

== See also ==
{{Portal|Mathematics}}
*[[Abstract nonsense]] 
*[[Glossary of category theory]]

{{Category theory}}
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