it is faithful; that is, for any pair of objects $x,y\in C$ the function $F: C(x,y) \to D(F x,F y)$ is injective, and

it is full on isomorphisms, meaning that for any pair of objects $x,y\in C$ the function $F: Iso_C(x,y) \to Iso_D(F x, F y)$ is surjective (hence bijective), where $Iso_C(x,y)$ means the set of isomorphisms from $x$ to $y$ in $C$.

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

Properties

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

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

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

A species is a functor from the category $Bij$ of finite sets and bijections to $Set$, and the functors that are obtained by taking left Kan extensions of species along the embedding $I:Bij \to Set$ are called analytic functors. Now taking left Kan extensions along $I$ 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 $A$ is uniquely determined by $FA$ up to unique isomorphism. However, we do not really need faithfulness for this; bijectivity on isos suffices.

Revised on April 13, 2016 13:09:48
by Mike Shulman
(192.195.154.58)