constant functor in nLab

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Definition

A constant functor Δd:C→D\Delta d: C\to D is a functor that maps each object of the category CC to a fixed object d∈Dd\in D and each morphism of CC to the identity morphism of that fixed object.

(The notation Δd\Delta d is suggested by the fact that if d:1→Dd: 1 \to D names an object dd of DD, then the composite

1→dD→ΔD C1 \stackrel{d}{\to} D \stackrel{\Delta}{\to} D^C

names a functor Δd:C→D\Delta d: C \to D which is the constant functor at dd. Here Δ:D→D C\Delta: D \to D^C denotes a diagonal functor.)

Viewing D≅[1,D]D \cong [1,D], it is possible to see that Δ\Delta is a base change of the functor categories D 1D^1 and D CD^C, given the functor ! C:C→1!_C: C \to 1.

More generally, we may say that a functor F:C→DF:C\to D is essentially constant if it is naturally isomorphic to a constant functor.

Properties

Note that a constant functor can be expressed as the composite

C→!1→[d]D.C \stackrel{!}{\to} 1 \stackrel{[d]}{\to} D.

Here 11 is a terminal category (exactly one object and exactly one morphism, namely the identity), and [d][d] denotes the unique functor from 11 with F(•)=dF(\bullet) = d and F(Id •)=Id dF(Id_\bullet) = Id_d. It follows that F:C→DF:C\to D is essentially constant if and only if it factors through 11 up to isomorphism.

If CC has at least one object, then there is a codescent object

C×C×C→→→C×C⇉C→1. C\times C\times C \underoverset{\to}{\to}{\to} C\times C \rightrightarrows C \to 1.

Therefore, in this case a functor F:C→DF:C\to D is essentially constant if and only if we have a natural isomorphism between the two composites C×C⇉C→FDC\times C \rightrightarrows C \xrightarrow{F} D (i.e. isomorphisms F(c 1)≅F(c 2)F(c_1)\cong F(c_2) natural in c 1,c 2∈Cc_1,c_2\in C) that satisfies a cocycle condition?. This is a categorified version of the statement that a function with inhabited domain is a constant function if and only if the images of any two elements are equal. Just as in that case, there is a distinction in the trivial case: the identity functor of the empty category satisfies this weaker condition, but is not essentially constant because there is no object for it to be constant at.

Examples

• For FF any functor, a natural transformation

  Δ d⇒F\Delta_d \Rightarrow F

  from a constant functor into FF is precisely a cone over FF. Similarly a natural transformation

  F⇒Δ dF \Rightarrow \Delta_d

  is a cocone.

• constant presheaf, locally constant sheaf
• diagonal functor
• constant morphism, constant function