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 component function $F \colon C(x,y) \to D(F x,F y)$ between hom-sets is injective, and

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

More generally, a 1-morphism $f \colon C\to D$ in any 2-category $K$ is called a pseudomonic morphism if the square analogous to that below is a 2-pullback, or equivalently if $K(X,C)\to K(X,D)$ is a pseudomonic functor for any $X$.

Properties

Every fully faithful functor is pseudomonic, and every pseudomonic functor is conservative, as well as essentially injective. In fact, being full on isomorphisms is exactly what essential injectivity means.

A functor $F \colon C \to D$ is pseudomonic if and only if the commuting square

$$\array{ C &\overset{Id}{\longrightarrow}& C \\ \Big\downarrow\mathrlap{^{Id}} && \Big\downarrow\mathrlap{{}^F} \\ C &\underset{F}{\longrightarrow}& D }$$

is a 2-pullback in Cat (cf. the pullback characterization of 1-monomorphisms, here).

Examples

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

A combinatorial species is a functor from the category $Bij$ of finite sets and bijections between them to Set, and the functors that are obtained by taking left Kan extensions of species along the subcategory-inclusion $I \colon Bij \to Set$ 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 $F A$ up to unique isomorphism. However, we do not really need faithfulness for this; bijectivity on isos suffices.

References

Formalization in cubical Agda:

• 1lab, Pseudomonic Functors