User:Fropuff/Drafts/Comma category

comma category (T ↓ S)

${\begin{matrix}T(\alpha )&{\xrightarrow {T(g)}}&T(\alpha ')\\f{\Bigg \downarrow }&&{\Bigg \downarrow }f'\\S(\beta )&{\xrightarrow[{S(h)}]{}}&S(\beta ')\end{matrix}}$

• 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
  • a universal morphism from U to A is a terminal object of the over category (U ↓ 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
  • a universal morphism from A to U is an initial object of the under category (A ↓ U)
  • coslice category, objects under A, written (A ↓ C) or A/C
  • (F ↓ Δ) category of cones from F

Slice category

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

• If I is an initial object in C then (I → A) is an initial object in C/A.
• The coproduct of f_X and f_Y is the natural morphism f_X+Y.

• (id_A : A → A) is a terminal object in C/A.
• Products in C/A are pullbacks in C.

Examples

• 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

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.

Examples

• 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