nLab
constant functor

Contents

Definition

A constant functor Δ(d):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.

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.

Examples

  • 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 are natural transformation

    FΔ dF \Rightarrow \Delta_d

    is a cocone.

Revised on June 20, 2013 12:47:01 by Urs Schreiber (82.169.65.155)