A functor is pseudomonic if
it is faithful; that is, for any pair of objects the component function between hom-sets is injective, and
it is full on isomorphisms, meaning that for any pair of objects the function to the set of isomorphisms between them is surjective (hence bijective), where .
More generally, a 1-morphism in any 2-category is called a pseudomonic morphism if the square analogous to that below is a 2-pullback, or equivalently if is a pseudomonic functor for any .
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 is pseudomonic if and only if the commuting square
is a 2-pullback in Cat (cf. the pullback characterization of 1-monomorphisms, here).
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 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 are called analytic functors. Now taking left Kan extensions along 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 is uniquely determined by up to unique isomorphism. However, we do not really need faithfulness for this; bijectivity on isos suffices.
Formalization in cubical Agda:
Last revised on September 22, 2023 at 12:06:06. See the history of this page for a list of all contributions to it.