nLab
proadjoint

Contents

Definition

Given a functor F:CDF: C\to D we say that FF admits a proadjoint if the canonical extension pro(F):pro(C)pro(D)pro(F): pro(C)\to pro(D) of FF to the categories of pro-objects has a left adjoint GG. In other words, there is a functor G:pro(D)pro(C)G: pro(D)\to pro(C) and a bijection

pro(C)(GY,X)pro(D)(Y,y(F)X) pro(C)(GY',X) \cong pro(D)(Y',y(F)X)

natural in XX and YY', where y:Cpro(C)y:C\hookrightarrow pro(C) is the Yoneda embedding into the category of proobjects pro(C)Set C oppro(C)\subset Set^{C^{op}}. Equivalently, for every prorepresentable functor X:C opSetX:C'^{op}\to Set, the functor XXFX\mapsto X\circ F is also prorepresentable.

References

  • J.-M. Cordier, T. Porter, Shape theory : Categorical Methods of Approximation, (sec. 2.3), Mathematics and its Applications, Ellis Horwood Ltd., March 1989, 207 pages.Dover addition (2008) (Link to publishers here)

Revised on April 30, 2011 17:44:17 by Urs Schreiber (89.204.153.106)