power object



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


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.


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