Constant FunctorEdit
Constant functor
In category theory, a constant functor is a simple yet useful construction that packages a fixed object in a target category into a functor from another category. Given a category C and a fixed object d in a category D, the constant functor Δ_d: C → D sends every object X ∈ ob(C) to d and every morphism f: X → Y to the identity morphism id_d. This makes Δ_d a bona fide functor, because it preserves identities and respects composition: Δ_d(id_X) = id_d and Δ_d(g ∘ f) = id_d = id_d ∘ id_d = Δ_d(g) ∘ Δ_d(f).
Intuitively, Δ_d is the way to “plug in” a constant object d into every position of a diagram indexed by C. It is a basic device for studying how a fixed object interacts with a whole diagram, and it plays a central role in the language of cones, limits, and related universal constructions.
Definition
- Let C be a category and D a category that contains a fixed object d. The constant functor Δ_d: C → D is defined by:
- For each object X in C, Δ_d(X) = d.
- For each morphism f: X → Y in C, Δ_d(f) = id_d.
- A natural transformation α: Δ_d ⇒ F, where F: C → D is another functor, consists of a family of morphisms α_X: d → F(X) for all X in C, compatible with the arrows of C. Concretely, for every f: X → Y in C, F(f) ∘ α_X = α_Y.
This compatibility condition means that such natural transformations are the same thing as cones with apex d over the diagram F. In other words, natural transformations from a constant functor to a diagram encode the way to funnel data from a fixed object into every stage of the diagram in a coherent way.
Basic properties
- Universality with cones: As noted, Hom-sets from Δ_d to F correspond to cones from d into F. This viewpoint is often used to relate constant functors to limits.
- Limits and cones: If F: J → D is a diagram, a limit of F can be characterized via a universal cone p_j: L → F(j). Natural transformations from Δ_d to F can be interpreted as cones from d to F, and the limit condition says every such cone factors uniquely through the limit cone.
- Dual perspective with colimits: Dually, a cocone from F to Δ_d corresponds to a natural transformation from F to a constant functor, tying constant functors to colimit-like constructions in a dual way.
- In common categories: In categories like Set, Grp, and Ab, the constant functor Δ_d behaves in ways familiar from set-theoretic or algebraic constructions: it simply places the fixed object d in every position of a diagram and uses the identity maps to connect those positions.
Examples
- Example in Set: Let C be any small category and D = Set with a fixed set S. The constant functor Δ_S: C → Set assigns S to every object and id_S to every morphism. A natural transformation Δ_S ⇒ F, for F: C → Set, is the same as a cone with apex S over the diagram F. Concretely, this means giving, for each object X of C, a function S → F(X) that is compatible with the action of F on morphisms.
- If F is a diagram indexed by a category with a single nontrivial arrow f: A → B, a natural transformation Δ_S ⇒ F is determined by a pair of functions S → F(A) and S → F(B) such that F(f) ∘ (S → F(A)) = (S → F(B)).
- Example in Ab: Take D = Ab and a fixed abelian group A. The constant functor Δ_A: C → Ab sends every object to A and every morphism to id_A. A natural transformation Δ_A ⇒ F corresponds to, for each object X, a homomorphism A → F(X) that commutes with the structure maps of F.
Connections to other concepts
- Relation to diagrams and cones: The notion of a cone over a diagram F is exactly the same data as a natural transformation Δ_d ⇒ F for some apex d.
- Role in universal properties: Since cones underpin the definition of limits, constant functors provide a natural framework for expressing when a diagram has a universal cone and how that cone universalizes maps from any fixed apex.
- Diagrammatic viewpoints: The constant functor is part of the broader vocabulary that translates between object-level data (an apex object) and diagrammatic coherence (a family of maps indexed by the diagram).