# Category:Category Theory

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This category contains results about Category Theory.

Definitions specific to this category can be found in Definitions/Category Theory.

**Category theory** is the branch of abstract algebra which studies categories.

It can be described as the **theory of functors**.

## Subcategories

This category has the following 19 subcategories, out of 19 total.

### A

- Abelian Categories (empty)

### C

### D

- Direct Sums (1 P)

### E

### F

- Free Monoids (2 P)
- Functors (empty)

### L

### M

### N

- Natural Transformations (empty)

### P

- Preadditive Categories (4 P)

### S

- Sheaves (empty)

### U

- Universal Properties (18 P)

## Pages in category "Category Theory"

The following 65 pages are in this category, out of 65 total.

### C

- Category Axioms are Self-Dual
- Category has Finite Limits iff Finite Products and Equalizers
- Category has Products and Equalizers iff Pullbacks and Terminal Object
- Category Induces Preorder
- Category of Pointed Sets is Category
- Cayley's Theorem (Category Theory)
- Cayley's Theorem (Category Theory)/Historical Note
- Characterization of Metacategory via Equations
- Coequalizer is Epimorphism
- Composite Functor is Functor
- Composition of Functors is Associative
- Composition with Zero Morphism is Zero Morphism
- Contravariant Hom Functor maps Colimits to Limits
- Covariant Hom Functor is Continuous
- Covariant Hom Functor is Functor

### E

### I

- Identity Functor is Functor
- Identity Functor is Left Identity
- Identity Functor is Right Identity
- Identity Morphism is Unique
- Identity Morphism of Product
- Initial Object is Unique
- Injective iff Projective in Dual Category
- Inverse Morphism is Unique
- Isomorphism (Category Theory) is Epic
- Isomorphism (Category Theory) is Monic