nLab
pseudomonic functor

Contents

Definition

A functor F:CDF: C \to D is pseudomonic if

  1. it is faithful; that is, for any pair of objects x,yCx,y\in C the function F:C(x,y)D(Fx,Fy)F: C(x,y) \to D(F x,F y) is injective, and
  2. it is full on isomorphisms, meaning that for any pair of objects x,yCx,y\in C the function F:Iso C(x,y)Iso D(Fx,Fy)F: Iso_C(x,y) \to Iso_D(F x, F y) is surjective (hence bijective), where Iso C(x,y)Iso_C(x,y) means the set of isomorphisms from xx to yy in CC.

More generally, a morphism f:CDf:C\to D in any 2-category KK is called pseudomonic morphism if the corresponding square is a pullback, or equivalently if K(X,C)K(X,D)K(X,C)\to K(X,D) is a pseudomonic functor for any XX.

Properties

Every full and faithful functor is pseudomonic, and every pseudomonic functor is conservative. A functor F:CDF: C \to D is pseudomonic if and only if the square

C Id C Id F C F D \array{ C &\stackrel{Id}{\to}& C \\ \downarrow^{Id} && \downarrow^F \\ C &\stackrel{F}{\to}& D }

is a pullback in Cat.

Examples

An interesting example of the notion appears in the context of Joyal’s species of structures.

A species is a functor from the category BijBij of finite sets and bijections to SetSet, and the functors that are obtained by taking left Kan extensions of species along the embedding I:BijSetI:Bij \to Set are called analytic functors. Now taking left Kan extensions along II is pseudomonic, and this implies that the coefficients of an analytic functor are unique up to isomorphism.

I think that in a sense pseudomonic functors are precisely the functors for which it makes sense to say that AA is uniquely determined by FAFA up to isomorphism (although we do not really need faithfulnes for this, bijectivity on isos suffices).

Revised on October 26, 2010 18:20:47 by Urs Schreiber (131.211.232.186)