We say that is a parametric right adjoint, or p.r.a., if the functor is a right adjoint.
Any polynomial functor is p.r.a., since then can be identified with , which has the left adjoint .
If is a presheaf category and is p.r.a., then the comma category (also called the Artin gluing in this context) is again a presheaf category. Conversely, if is a presheaf category, then preserves connected limits, and thus is p.r.a. if it is accessible.
Central to the theory of parametric right adjoints is the notion of -generic morphisms. For any functor , a morphism is (strictly) -generic if any commutative square of the following form:
has a unique filler . A generic factorization of a map is a factorization
such that is -generic. Note that by the definition of genericity, generic factorizations are unique whenever they exist. If is a monad and any map has a generic factorization, then there is an induced orthogonal factorization system on the Kleisli category of in which -generic maps are the left class and the right class are the “free” maps, i.e. those which factor through the unit of .
A functor is a parametric right adjoint iff every map has a generic factorization.
This is Proposition 2.6 of “Familial 2-functors and parametric right adjoints.”
P.r.a. functors between presheaf categories have an especially nice form.
A functor between presheaf categories is p.r.a. iff any map has a generic factorization, where is the representable presheaf on an object .
This is Proposition 2.10 of “Familial 2-functors and parametric right adjoints.” The “only if” direction is the previous proposition, while for the “if” direction, the given hypothesis allows us to define the functor
sending an object to the object occurring in its generic factorization. Note that is equivalently the opposite of the category of elements of . The definition of genericity, along with the Yoneda lemma, then shows that
which preserves connected limits, since it is a coproduct of representables.
In particular, a p.r.a. functor is determined by giving the object together with the functor . We can think of as the setof all possible “shapes” which allows us to “glue together” to obtain an element of shape , and as specifying exactly what each of those shapes looks like. Then the above formula for says that we look at all possible shapes we can glue to get something of shape , and for each such we look at all the “diagrams” in of the corresponding shape .
Consider the free category monad on the category of quivers, such that is the quiver with the same objects as and whose arrows are finite composable strings of arrows in .. Here is the monoid regarded as a one-object category, and thus an object of is a quiver together with a natural number assigned to each edge. For any quiver , the natural augmentation assigns to each composable string of arrows its length.
The left adjoint of this functor takes as input a quiver with natural number “lengths” assigned to each of its arrows, and creates a new quiver by gluing together a copy of the quiver (with no arrows other than those drawn) for each arrow of “length” . Thus is a parametric right adjoint.
is of course a presheaf category , where is the category . The category , i.e. the opposite of the category of elements of , has objects and nonidentity arrows for all . Finally, the functor sends to the quiver with one object and no arrows, and to the quiver described above.