nLab pseudomonic functor




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 square analogous to that below 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.


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.


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.

Arguably, pseudomonic functors are precisely the functors for which it makes sense to say that AA is uniquely determined by FAF A up to unique isomorphism. However, we do not really need faithfulness for this; bijectivity on isos suffices.

Last revised on June 3, 2019 at 07:18:44. See the history of this page for a list of all contributions to it.