User:Fropuff/Drafts/Comma category
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comma category (T ↓ S)
- hom-set category (A ↓ B) = Hom(A, B) as a discrete category
- morphism (or arrow) category (C ↓ C) = C2
- (U ↓ A), objects U over A, or morphisms from U to A
- slice category, objects over A, written (C ↓ A) or C/A
- (Δ ↓ F) category of cones to F
- (A ↓ U), objects U under A, or morphisms from A to U
- coslice category, objects under A, written (A ↓ C) or A/C
- (F ↓ Δ) category of cones from F
Slice category
[edit]Let C be a category and let A be an object in C. The slice category is denoted (C ↓ A) or C/A.
- objects are morphisms to A in C, e.g. f : X → A
- morphisms are commutative triangles φ : (f : X → A) → (g : Y → A) with f = g∘φ
The forgetful functor, U : C/A → C, assigns to each morphism f : X → A its domain X. If C has finite products this functor has a right-adjoint which assigns to each space Y the projection map (A × Y → A). U then commutes with colimits.
Limits and colimits
[edit]- If I is an initial object in C then (I → A) is an initial object in C/A.
- The coproduct of fX and fY is the natural morphism fX+Y.
- (idA : A → A) is a terminal object in C/A.
- Products in C/A are pullbacks in C.
Examples
[edit]- If A is terminal, then C/A is isomorphic to C.
- If C is a poset category, C/A is the principal ideal of objects less than A.
- Set/ℕ is the category of graded sets (morphisms must preserve the grade, so perhaps different than a multiset)
Coslice category
[edit]Let C be a category and let A be an object in C. The coslice category is denoted (A ↓ C) or A/C.
- objects are morphisms from A in C, e.g. f : A → X
- morphisms are commutative triangles φ : (f : A → X) → (g : A → Y) with g = φ∘f.
Limits and colimits
[edit]Examples
[edit]- If A is initial, then A/C is isomorphic to C.
- •/Set is the category of pointed sets
- •/Top is the category of pointed spaces
- R/CRing is the category of commutative R-algebras