constant functor



A constant functor Δd:CD\Delta d: C\to D is a functor that maps each object of the category CC to a fixed object dDd\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:1Dd: 1 \to D names an object dd of DD, then the composite

1dDΔD C1 \stackrel{d}{\to} D \stackrel{\Delta}{\to} D^C

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

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


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:CDF: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×CC×CC1. C\times C\times C \underoverset{\to}{\to}{\to} C\times C \rightrightarrows C \to 1.

Therefore, in this case a functor F:CDF:C\to D is essentially constant if and only if we have a natural isomorphism between the two composites C×CCFDC\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 2Cc_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.


  • For FF any functor, a natural transformation

    Δ dF\Delta_d \Rightarrow F

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

    FΔ dF \Rightarrow \Delta_d

    is a cocone.

Revised on July 1, 2017 20:58:09 by Todd Trimble (