Category Theory

Contents
Categories and Precategories
Precategory
Isomorphism

Homotopy Equivalence
Lemma
Identity to Isomorphism
Example
Category

Example
Lemma
Isomorphism to Identity
Lemma
Example (Preorder)
Example
Groupoid
Fundamental Pregroupoid
Homotopy Precategory of Types
Example
Functor



Functor Precategory
Lemma
A natural transformation Ξ³ : F β G is an isomorphism in B^A if and only if each Ξ³_a is an isomorphism in B.
Theorem
Composite of functors
Definition
Definition
Lemma
Lemma
Adjoint Functors/Adjunction

Functor F : A β B β is-left-adjoint(F) β‘
- G : B β A,
- Ξ· : 1_A β G F (unit),
- Ξ΅ : F G β 1_B (counit),
- (e F) (F Ξ·) = 1_F, (G Ξ΅) (Ξ· G) = 1_G (triangle identities).
Equivalence
Definition
Lemma
Injective
'Faithful'.
Surjective
'Full'.
Split Essentially Surjective
Lemma
Essentially Surjective
Weak Equivalence
Lemma
Isomorphism
F is bijective and F_0 is an equivalence of types. A β
B.
Lemma
Lemma
Lemma
Dual

Product
Lemma
Yoneda Embedding
Yoneda Lemma

Corollary
Corollary
Representable
Lemma
Strict Category

Monomorphism

β -category
Structure Identity Principle
Notion of Structure
Theorem: Structure identity principle
Definition
Rezk Completion
Lemma
Theorem
Theorem
Diagonal Functor

Functor Category

Diagram



Commutative Diagram

Groupoid

Internal Category

Small category

Initial Object
Terminal Object
Category of Sets

Category of Groups

Category of Rings
Category of Modules

Category of Manifolds

Categorification

Decategorification

Indexed category

Limit

Coinduction

Corecursion

References