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User:Fropuff/Drafts/Comma category

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comma category (TS)

  • hom-set category (AB) = Hom(A, B) as a discrete category
  • morphism (or arrow) category (CC) = C2
  • (UA), objects U over A, or morphisms from U to A
    • slice category, objects over A, written (CA) or C/A
    • (Δ ↓ F) category of cones to F
  • (AU), objects U under A, or morphisms from A to U
    • coslice category, objects under A, written (AC) or A/C
    • (F ↓ Δ) category of cones from F

Slice category

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Let C be a category and let A be an object in C. The slice category is denoted (CA) or C/A.

  • objects are morphisms to A in C, e.g. f : XA
  • morphisms are commutative triangles φ : (f : XA) → (g : YA) with f = g∘φ

The forgetful functor, U : C/AC, assigns to each morphism f : XA its domain X. If C has finite products this functor has a right-adjoint which assigns to each space Y the projection map (A × YA). U then commutes with colimits.

Limits and colimits

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  • If I is an initial object in C then (IA) is an initial object in C/A.
  • The coproduct of fX and fY is the natural morphism fX+Y.
  • (idA : AA) is a terminal object in C/A.
  • Products in C/A are pullbacks in C.

Examples

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  • 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

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Let C be a category and let A be an object in C. The coslice category is denoted (AC) or A/C.

  • objects are morphisms from A in C, e.g. f : AX
  • morphisms are commutative triangles φ : (f : AX) → (g : AY) with g = φ∘f.

Limits and colimits

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Examples

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