Constant Functor

definition example

For an object $c$ of $\mathcal{D}$, the constant functor ${\Delta }_c:\mathcal{J}\to \mathcal{D}$ sends every object of $\mathcal{J}$ to $c$ and every morphism to ${\mathrm{id}}_c$. In Haskell it is Const c (functorial in its second argument, DaoFP Exercise 8.3.3).

Sources: DaoFP §8.2, §8.3, §9.4 (“Picking objects”), §10.2 (“The diagonal functor”); 7 Sketches Definition 3.92 (cones); Kittenlab Lecture 9; CTfS Exercise 4.3.1.10, §5.1.2

• Picking an object of $\mathcal{C}$ is the same as a functor $\underline{1}\to \mathcal{C}$, or a constant functor from any $\mathcal{J}$; picking a pair, a functor from the discrete $2$; picking an arrow, a functor from the Walking Arrow.
• A Cone over a Diagram $D:\mathcal{J}\to \mathcal{C}$ with apex $x$ is a Natural Transformation ${\Delta }_x\Rightarrow D$; a Cocone is $D\Rightarrow {\Delta }_x$ (DaoFP §9.5, Kittenlab Lecture 9: ”$\Delta :\mathcal{C}\to {\mathcal{C}}^{\mathsf{D}}$ sends $X$ to the constant functor at $X$”). Hence limits and colimits are adjoints to ${\Delta }_{(-)}:\mathcal{C}\to [\mathcal{J},\mathcal{C}]$: $\mathrm{Colim}⊣\Delta ⊣\mathrm{Lim}$.
• A function $f:X\to Y$ gives a natural transformation between constant functors ${\Delta }_X\Rightarrow {\Delta }_Y:\mathcal{J}\to \mathbf{Set}$ whose every component is $f$; naturality squares commute trivially because the functors send all arrows to identities (CTfS Exercise 4.3.1.10). So $\Delta :\mathcal{C}\to [\mathcal{J},\mathcal{C}]$ is itself a functor, and CTfS §5.1.2 obtains limits and colimits of all $\mathcal{J}$-shaped diagrams at once as its adjoints ${\Sigma }_t⊣{\Delta }_t⊣{\Pi }_t$.
• The Diagonal Functor $\Delta :\mathcal{C}\to \mathcal{C}\times \mathcal{C}$ is the constant functor curried: $\mathcal{C}\times \mathcal{C}\cong [2,\mathcal{C}]$, “which is why we use the same symbol $\Delta$ for both”.
• A constant $\mathbf{Set}$-valued functor is the most lossy model of a category (DaoFP §9.6); it is represented by the Initial Object when the constant is $1$ (DaoFP Exercise 9.8.4; Kittenlab Lecture 10: ${\mathbb{R}}^0$ represents the constant singleton functor on $\mathbf{Vect}$).

JuliaLeanHaskell

Docs: FinCats · Categories & functors — Kittenlab Lecture 9, Lecture 10

Builds on: Category (Category), Functor (Functor) — run those notes’ Julia code first.

struct ConstFunctor{C<:Category, D<:Category, Ob, Hom} <: Functor{C, D}
  d::D; c::Ob
end
ob_map(F::ConstFunctor, _) = F.c
hom_map(F::ConstFunctor, _) = id(F.d, F.c)

#check CategoryTheory.Functor.const     -- (const J).obj X : J ⥤ C, the constant functor at X
#check CategoryTheory.Limits.Cone       -- structure Cone F: pt, π : (const J).obj pt ⟶ F

data Const c a = Const c
instance Functor (Const c) where
  fmap _ (Const c) = Const c