Category Theory Symbols with Examples
April 14, 2025•651 words
Category Theory Symbols with Examples
1. Objects and Morphisms
| Symbol | Name | Meaning | Example |
|---|---|---|---|
| A, B, C | Objects | Abstract elements in a category | A, B ∈ Ob(C) |
| f, g | Morphisms | Arrows between objects | f: A → B |
| idₐ | Identity | Identity morphism on A | idₐ: A → A |
| ∘ | Composition | Compose morphisms | g ∘ f: A → C (if f: A→B, g: B→C) |
2. Categories and Collections
| Symbol | Name | Meaning | Example |
|---|---|---|---|
| 𝒞, 𝒟, 𝐄 | Category names | Categories are collections of morphisms | 𝒞 = Set, 𝒟 = Grp |
| Ob(𝒞) | Objects | Set of objects in category 𝒞 | A ∈ Ob(𝒞) |
| Hom(A, B) | Hom-set | Set of morphisms A → B in 𝒞 | f ∈ Hom(A, B) |
| Mor(𝒞) | Morphisms | Collection of all morphisms | Mor(Set) = all functions |
3. Functors
| Symbol | Name | Meaning | Example |
|---|---|---|---|
| F, G | Functors | Structure-preserving map between categories | F: 𝒞 → 𝒟 |
| F(f) | Action on morphisms | Maps f: A→B in 𝒞 to F(f): F(A)→F(B) | F(f): F(A) → F(B) |
4. Natural Transformations
| Symbol | Name | Meaning | Example |
|---|---|---|---|
| η, α | Natural transformation | η: F ⇒ G maps functors | ηₐ: F(A) → G(A) (component at A) |
| ⇒ | Natural transformation | Arrow between functors | η: F ⇒ G |
| ηₐ | Component | Morphism for object A | ηₐ: F(A) → G(A) |
5. Commutative Diagrams
| Symbol | Name | Meaning | Example |
|---|---|---|---|
| →, ↦ | Arrow | Represents morphisms | f: A → B |
| ⇨ | Double arrow | Natural transformation or functor map | η: F ⇒ G |
| ⬚ | Commutativity | Diagram commutes | g ∘ f = h |
6. Products and Coproducts
| Symbol | Name | Meaning | Example |
|---|---|---|---|
| × | Product | Product of two objects | A × B |
| π₁, π₂ | Projections | Morphisms from product | π₁: A×B → A, π₂: A×B → B |
| ⊕ or + | Coproduct | Sum object in a category | A + B (in Set: disjoint union) |
| ι₁, ι₂ | Injections | Morphisms into coproduct | ι₁: A → A⊕B, ι₂: B → A⊕B |
7. Limits and Colimits
| Symbol | Name | Meaning | Example |
|---|---|---|---|
| lim← | Limit (projective) | Cone over diagram D | lim← D |
| colim→ | Colimit (inductive) | Cocone over diagram D | colim→ D |
| Δ | Constant diagram | Functor into diagram shape | ΔA: Constant diagram at A |
| cone | Universal cone | Limit property | u: X → Aᵢ commutes for all i |
8. Special Categories and Functors
| Symbol | Name | Meaning | Example |
|---|---|---|---|
| 1 | Terminal object | Unique morphism from any A → 1 | ∀A, ∃! f: A → 1 |
| 0 | Initial object | Unique morphism from 0 → A | ∀A, ∃! f: 0 → A |
| Opp(𝒞) | Opposite category | Reverses morphisms | Hom{𝒞ᵒᵖ}(A,B) = Hom𝒞(B,A) |
| Yoneda | Yoneda embedding | Hom functor mapping objects to sets | hᴀ: C ↦ Hom(–, A) |
9. Higher Categories / Meta-notation
| Symbol | Name | Meaning | Example |
|---|---|---|---|
| 2-𝒞 | 2-Category | Categories with functors and transformations | Cat: category of small categories |
| ∞-𝒞 | ∞-Category | Higher-order category | ∞-Topos |