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Definition

Let $T:A\to B$ be a functor such that $A$ has a terminal object $1$. Then $T$ can be factored as the composite

We say that $T$ is a parametric right adjoint, or p.r.a., if the functor $T_1$ is a right adjoint.

A monad is called p.r.a. if its functor part is p.r.a. and moreover its unit and multiplication are cartesian.

Properties

• Since ${\Sigma }_{T1}$ creates connected limits, if $T$ is p.r.a. then it preserves connected limits, and in particular preserves pullbacks. It follows that any p.r.a. monad is a cartesian monad.

• Conversely, an accessible functor between presheaf categories is p.r.a. if and only if it preserves connected limits.

• Any polynomial functor ${\Sigma }_h{\Pi }_g{f}^{*}$ is p.r.a., since then $T_1$ can be identified with ${\Pi }_g{f}^{*}$, which has the left adjoint ${\Sigma }_f{g}^{*}$.

• If $E$ is a presheaf category and $T:E\to \mathrm{Set}$ is p.r.a., then the comma category $\mathrm{Set}/T$ (also called the Artin gluing in this context) is again a presheaf category. Conversely, if $\mathrm{Set}/T$ is a presheaf category, then $T$ preserves connected limits, and thus is p.r.a. if it is accessible.

Generic morphisms

Central to the theory of parametric right adjoints is the notion of $T$-generic morphisms. For any functor $T$, a morphism $f:B\to TA$ is (strictly) $T$-generic if any commutative square of the following form:

has a unique filler $TA\to TX$. A generic factorization of a map $f:B\to TA$ is a factorization

such that $g$ is $T$-generic. Note that by the definition of genericity, generic factorizations are unique whenever they exist. If $T$ is a monad and any map $B\to TA$ has a generic factorization, then there is an induced orthogonal factorization system on the Kleisli category of $T$ in which $T$-generic maps are the left class and the right class are the “free” maps, i.e. those which factor through the unit of $T$.

P.r.a. functors between presheaf categories have an especially nice form.

In particular, a p.r.a. functor $T:[I^{\mathrm{op}},\mathrm{Set}]\to [J^{\mathrm{op}},\mathrm{Set}]$ is determined by giving the object $T1\in [J^{\mathrm{op}},\mathrm{Set}]$ together with the functor $E_T:y/T1=\mathrm{el}(T1{)}^{\mathrm{op}}\to [I^{\mathrm{op}},\mathrm{Set}]$. We can think of $T1(j)$ as the setof all possible “shapes” which $T$ allows us to “glue together” to obtain an element of shape $j$, and $E_T$ as specifying exactly what each of those shapes looks like. Then the above formula for $T(Z)(j)$ says that we look at all possible shapes $x\in T1(j)$ we can glue to get something of shape $j$, and for each such $x$ we look at all the “diagrams” in $Z$ of the corresponding shape $E_T(x)$.

Examples

Free categories

Consider the free category monad $T$ on the category $\mathrm{Quiv}$ of quivers, such that $TA$ is the quiver with the same objects as $A$ and whose arrows are finite composable strings of arrows in $A$.. Here $T1$ is the monoid $\mathbb{N}$ regarded as a one-object category, and thus an object of $\mathrm{Quiv}/T1$ is a quiver together with a natural number assigned to each edge. For any quiver $A$, the natural augmentation $TA\to T1$ assigns to each composable string of arrows its length.

The left adjoint of this functor $T_1:\mathrm{Quiv}\to \mathrm{Quiv}/T1$ 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 $[n]=(0\to 1\to \dots \to n)$ (with no arrows other than those drawn) for each arrow of “length” $n$. Thus $T$ is a parametric right adjoint.

$\mathrm{Quiv}$ is of course a presheaf category $[Q^{\mathrm{op}},\mathrm{Set}]$, where $Q$ is the category $0\rightrightarrows 1$. The category $y/T1$, i.e. the opposite of the category of elements of $T1$, has objects $\mathbb{N}\bigsqcup \{\perp \}$ and nonidentity arrows $\perp \rightrightarrows n$ for all $n\in \mathbb{N}$. Finally, the functor $E_T:y/T1\to \mathrm{Quiv}$ sends $\perp$ to the quiver with one object and no arrows, and $n$ to the quiver $[n]=(0\to 1\to \dots \to n)$ described above.

References

• Aurelio Carboni and Peter Johnstone, Connected limits, familial representability and Artin glueing, MR

• Mark Weber, Generic morphisms, parametric representations, and weakly cartesian monads, Theory and applications of categories, 13:191–234, 2004. link

• Mark Weber, Familial 2-functors and parametric right adjoints, 2008 link