nLab
pseudomonic functor

Definition

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

1. it is faithful; that is, for any pair of objects $x,y\in C$ the function $F:C(x,y)\to D(Fx,Fy)$ is injective, and
2. it is full on isomorphisms, meaning that for any pair of objects $x,y\in C$ the function $F:{\mathrm{Iso}}_C(x,y)\to {\mathrm{Iso}}_D(Fx,Fy)$ is surjective (hence bijective), where ${\mathrm{Iso}}_C(x,y)$ 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

$$\begin{array}{ccc}C& \stackrel{\mathrm{Id}}{\to }& C\\ {\downarrow }^{\mathrm{Id}}& & {\downarrow }^F\\ C& \stackrel{F}{\to }& D\end{array}$$\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 $\mathrm{Bij}$ of finite sets and bijections to $\mathrm{Set}$, and the functors that are obtained by taking left Kan extensions of species along the embedding $I:\mathrm{Bij}\to \mathrm{Set}$ 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.

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