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Contents 1 Hom functors 1.1 Properties 1.1.1 Preservation of limits 1.2 Categorical objects associated with Hom sets 2 Comma categories 2.1 Slice category 2.1.1 Limits and colimits 2.1.2 Examples 2.2 Coslice category 2.2.1 Limits and colimits 2.2.2 Examples

[edit] Hom functors

[edit] Properties

[edit] Preservation of limits

Covariant Hom functors preserve all limits. In particular, they preserve all small limits, and are therefore continuous. By duality, the contravariant Hom functors take colimits to limits. Covariant Hom functors do not necessarily preserve colimits.

Given a diagram F : J → C and an object X of C the limit of composite functor Hom(X, F–) : J → Set is given by the set of all cones from X to F:

lim Hom(X, F–) = Cone(X, F)

The limiting cone is given by the maps

where π_i(ψ) = ψ_i. If F has a limit in C then Hom(X, lim F) is naturally isomorphic to the set of all cones from X to F so that

Hom(X, lim F) = lim Hom(X, F–)

Moreover, the Hom functor Hom(X, –) takes the limiting cone of F to the limiting cone of Hom(X, F–). It follows that Hom(X, –) preserves the limits of F.

[edit] Categorical objects associated with Hom sets

The are great variety of objects associated with Hom sets. These are summarized in the following table. In this table

• C is a category,
• A, A₁, A₂, and B, B₁, B₂ are objects in C,
• and are morphisms in C.

Object	Type	Domain and range	Definition
	set
	function
	function
	function
	functor
	functor
	natural transformation
	natural transformation
	bifunctor
	functor
	functor

[edit] Comma categories

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

[edit] 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.

[edit] 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.

[edit] 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)

[edit] 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.

[edit] Limits and colimits

[edit] 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