Category Theory
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
Functors and Transformations
Functor
Natural Transformation
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
Composite of natural transformations and functors
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