Contents

category theory

# Contents

## Definition

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

$A \overset{T_1}{\to} B/T1 \overset{\Sigma_{T1}}{\to} B$

of $T$ applied to the slice category $A \simeq A/1$, followed by dependent sum (projection on the source).

We say that $T$ is a parametric right adjoint, or p.r.a., if the functor $T_1$ is a right adjoint. Parametric right adjoints are also called local right adjoints.

A monad is called p.r.a. if its functor part is p.r.a. and moreover its unit and multiplication are cartesian. Thus in particular it is a cartesian monad. A p.r.a. monad is also called a strongly cartesian monad.

## 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, a functor between presheaf categories is p.r.a. if it preserves connected limits. The reason is that (in the notation above), if $T$ preserves connected limits, then $T_1$ preserves all small limits, and a limit-preserving functor from a cototal category such as a presheaf category to a locally small category is a right adjoint.

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

• A parametric right adjoint functor (with locally small codomain) has in particular a left multi-adjoint, which sends each object $b\in B$ to the family of all units $\eta_{b,i} : b \to T L(b,i)$, where $i$ ranges over all morphisms $b\to T 1$ and $L : B/T1 \to A$ is the left adjoint of $T_1$. This is because any morphism $b\to T a$ induces a unique composite $i:b \to T a \to T 1$, and hence a unique factorization through $L(b,i)$. Conversely, if $A$ has a terminal object and $T:A\to B$ has a left multi-adjoint, then it is a parametric right adjoint.

## Generic morphisms

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

$\array{B & \overset{\alpha}{\to} &T X \\ ^f\downarrow && \downarrow^{T\gamma}\\ T A& \underset{T \beta}{\to} & T Z}$

has a unique filler of the form $T\delta : T A \to T X$. A generic factorization of a map $f\colon B\to T A$ is a factorization

$B \overset{g}{\to} T D \overset{T h}{\to} T A$

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 $B \to T A$ 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 $T$.

###### Proposition

A functor $T$ is a parametric right adjoint iff every map $B\to T A$ has a generic factorization.

###### Proof

This is Proposition 2.6 of (Weber08). In fact, the generic factorizations are precisely the universal maps in the left multi-adjoint of $F$ mentioned above.

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

###### Proposition

A functor $T\colon [I^{op},Set] \to [J^{op},Set]$ between presheaf categories is p.r.a. iff any map $y(j)\to T 1$ has a generic factorization, where $y(j)$ is the representable presheaf on an object $j\in J$.

###### Proof

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

$E_T \colon y/T1 \to [I^{op},Set]$

sending an object $(y(j) \to T 1)$ to the object occurring in its generic factorization. Note that $y/T1$ is equivalently the opposite of the category of elements of $T1$. The definition of genericity, along with the Yoneda lemma, then shows that

$T(Z)(j) = \coprod_{x\in T1(j)} [I^{op},Set](E_T(x),Z)$

which preserves connected limits, since it is a coproduct of representables.

In particular, a p.r.a. functor $T\colon [I^{op},Set] \to [J^{op},Set]$ is determined by giving the object $T1\in [J^{op},Set]$ together with the functor $E_T\colon y/T1 = el(T1)^{op} \to [I^{op},Set]$. 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 $E_T(x)$.

We can extract from this a description that is clearly a generalization of a polynomial functor.

###### Proposition

A functor $T\colon [I^{op},Set] \to [J^{op},Set]$ between presheaf categories is p.r.a. iff when expressed in terms of discrete fibrations, it is the composite

$DFib/I \xrightarrow{d^*} DFib/E \xrightarrow{c_*} DFib/K \xrightarrow{p_!} DFib/J$

for a polynomial in $Cat$

$I \xleftarrow{d} E \xrightarrow{c} K \xrightarrow{p} J$

where $p$ is a discrete fibration and $(d,c)$ is a two-sided discrete fibration (with in particular $d$ a fibration and $c$ an opfibration).

###### Proof

Let $p$ be the Grothendieck construction of $T1$, so that $K = el(T1)^{op}$, and $(d,c)$ the two-sided Grothendieck construction of $E_T\colon el(T1)^{op} \to [I^{op},Set]$ regarded as a profunctor from $K$ to $I$. The above formula tells us that $T = Lan_p \circ Hom(E_T,-)$, and when rewritten in terms of discrete fibrations this gives the above formula. More details are in Remark 2.12 of (Weber08).

That is, a p.r.a. functor between presheaf categories is the restriction to discrete fibrations of a certain kind of polynomial functor between slices of $Cat$. When $I$ and $J$ are discrete categories, then so are $K$ and $E$, so that p.r.a. functors between presheaf categories are a direct generalization of polynomial functors between slices of $Set$. But on the other hand, we can also say that polynomial functors between slices of Cat are a direct generalization of p.r.a. functors between presheaf categories.

## Examples

### Free categories

Consider the free category monad $T$ on the category $Quiv$ of quivers, such that $T A$ is the quiver with the same objects as $A$ and whose arrows are finite composable strings of arrows in $A$.. Here $T 1$ is the monoid $\mathbb{N}$ regarded as a one-object category, and thus an object of $Quiv/T1$ is a quiver together with a natural number assigned to each edge. For any quiver $A$, the natural augmentation $T A \to T 1$ assigns to each composable string of arrows its length.

The left adjoint of this functor $T_1\colon Quiv \to Quiv/T1$ 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.

$Quiv$ is of course a presheaf category $[Q^{op},Set]$, 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 $\mathbb{N} \sqcup \{\bot\}$ and nonidentity arrows $\bot \rightrightarrows n$ for all $n\in\mathbb{N}$. Finally, the functor $E_T \colon y/T1 \to Quiv$ sends $\bot$ to the quiver with one object and no arrows, and $n$ to the quiver $[n] = (0 \to 1 \to\dots \to n)$ described above.

• 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

In database theory p.r.a.s between copresheaf categories, known as data migration functor, are treated in