nLab
power object

Contents

Idea

The notion of power object generalizes the notion of power set from the category Set to an arbitrary category with finite limits.

Definition

Let CC be a category with finite limits. A power object of an object cCc \in C is

  • an object Ω c\Omega^c

  • a monomorphism cc×Ω c\in_c \hookrightarrow c \times \Omega^c

such that

  • for every other object dd and every monomorphism rc×dr \hookrightarrow c \times d there is a unique morphism χ r:dΩ c\chi_r : d \to \Omega^c such that rr is the pullback
r c c×d Id c×χ r c×Ω c \array{ r &\to& \in_c \\ \downarrow && \downarrow \\ c \times d &\stackrel{Id_c \times \chi_r}{\to}& c \times \Omega^c }

If CC may lack some finite limits, then we may weaken that condition as follows:

  • If CC has all pullbacks (but may lack products), then equip each of c\in_c and rr with a jointly monic pair of morphisms, one to cc and one to Ω c\Omega^c or dd, in place of the single monomorphism to the product of these targets; rr must then be the joint pullback

    r d χ r c c Ω c Id c c \array { r & \rightarrow & d \\ \downarrow & \searrow & & \searrow^{\chi_r} \\ c & & \in_c & \rightarrow & \Omega^c \\ & \searrow^{Id_c} & \downarrow \\ & & c }
  • If CC may lack some pullbacks, then we simply require that the pullback that rr is to equal must exist. But arguably we should require, if Ω c\Omega^c is to be a power object, that this pullback exists for any given map χ:dΩ c\chi: d \to \Omega^c.

Examples

Revised on April 20, 2011 13:05:24 by Urs Schreiber (131.211.233.58)