nLab
parametric right adjoint

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Contents

Definition

Let T:ABT\colon A\to B be a functor such that AA has a terminal object 11. Then TT can canonically be factored as the composite

AT 1B/T1Σ T1B A \overset{T_1}{\to} B/T1 \overset{\Sigma_{T1}}{\to} B

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

We say that TT is a parametric right adjoint, or p.r.a., if the functor T 1T_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 Σ T1\Sigma_{T1} creates connected limits, if TT 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 TT preserves connected limits, then T 1T_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 Σ hΠ gf *\Sigma_h \Pi_g f^* is p.r.a., since then T 1T_1 can be identified with Π gf *\Pi_g f^*, which has the left adjoint Σ fg *\Sigma_f \; g^*.

  • If EE is a presheaf category and T:ESetT\colon E \to Set is p.r.a., then the comma category Set/TSet/T (also called the Artin gluing in this context) is again a presheaf category. Conversely, if Set/TSet/T is a presheaf category, then TT 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 bBb\in B to the family of all units η b,i:bTL(b,i)\eta_{b,i} : b \to T L(b,i), where ii ranges over all morphisms bT1b\to T 1 and L:B/T1AL : B/T1 \to A is the left adjoint of T 1T_1. This is because any morphism bTab\to T a induces a unique composite i:bTaT1i:b \to T a \to T 1, and hence a unique factorization through L(b,i)L(b,i). Conversely, if AA has a terminal object and T:ABT: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 TT-generic morphisms. For any functor TT, a morphism f:BTAf\colon B\to T A is (strictly) TT-generic if any commutative square of the following form:

B α TX f Tγ TA Tβ TZ\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 Tδ:TATXT\delta : T A \to T X. A generic factorization of a map f:BTAf\colon B\to T A is a factorization

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

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

Proposition

A functor TT is a parametric right adjoint iff every map BTAB\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 FF mentioned above.

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

Proposition

A functor T:[I op,Set][J op,Set]T\colon [I^{op},Set] \to [J^{op},Set] between presheaf categories is p.r.a. iff any map y(j)T1y(j)\to T 1 has a generic factorization, where y(j)y(j) is the representable presheaf on an object jJj\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:y/T1[I op,Set] E_T \colon y/T1 \to [I^{op},Set]

sending an object (y(j)T1)(y(j) \to T 1) to the object occurring in its generic factorization. Note that y/T1y/T1 is equivalently the opposite of the category of elements of T1T1. The definition of genericity, along with the Yoneda lemma, then shows that

T(Z)(j)= xT1(j)[I op,Set](E T(x),Z) 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:[I op,Set][J op,Set]T\colon [I^{op},Set] \to [J^{op},Set] is determined by giving the object T1[J op,Set]T1\in [J^{op},Set] together with the functor E T:y/T1=el(T1) op[I op,Set]E_T\colon y/T1 = el(T1)^{op} \to [I^{op},Set]. We can think of T1(j)T1(j) as the setof all possible “shapes” which TT allows us to “glue together” to obtain an element of shape jj, and E TE_T as specifying exactly what each of those shapes looks like. Then the above formula for T(Z)(j)T(Z)(j) says that we look at all possible shapes xT1(j)x\in T1(j) we can glue to get something of shape jj, and for each such xx we look at all the “diagrams” in ZZ of the corresponding shape E T(x)E_T(x).

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

Proposition

A functor T:[I op,Set][J op,Set]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/Id *DFib/Ec *DFib/Kp !DFib/J DFib/I \xrightarrow{d^*} DFib/E \xrightarrow{c_*} DFib/K \xrightarrow{p_!} DFib/J

for a polynomial in CatCat

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

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

Proof

Let pp be the Grothendieck construction of T1T1, so that K=el(T1) opK = el(T1)^{op}, and (d,c)(d,c) the two-sided Grothendieck construction of E T:el(T1) op[I op,Set]E_T\colon el(T1)^{op} \to [I^{op},Set] regarded as a profunctor from KK to II. The above formula tells us that T=Lan pHom(E T,)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 CatCat. When II and JJ are discrete categories, then so are KK and EE, so that p.r.a. functors between presheaf categories are a direct generalization of polynomial functors between slices of SetSet. 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 TT on the category QuivQuiv of quivers, such that TAT A is the quiver with the same objects as AA and whose arrows are finite composable strings of arrows in AA.. Here T1T 1 is the monoid \mathbb{N} regarded as a one-object category, and thus an object of Quiv/T1Quiv/T1 is a quiver together with a natural number assigned to each edge. For any quiver AA, the natural augmentation TAT1T A \to T 1 assigns to each composable string of arrows its length.

The left adjoint of this functor T 1:QuivQuiv/T1T_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]=(01n)[n] = (0 \to 1 \to\dots \to n) (with no arrows other than those drawn) for each arrow of “length” nn. Thus TT is a parametric right adjoint.

QuivQuiv is of course a presheaf category [Q op,Set][Q^{op},Set], where QQ is the category 010 \rightrightarrows 1. The category y/T1y/T1, i.e. the opposite of the category of elements of T1T1, has objects {}\mathbb{N} \sqcup \{\bot\} and nonidentity arrows n\bot \rightrightarrows n for all nn\in\mathbb{N}. Finally, the functor E T:y/T1QuivE_T \colon y/T1 \to Quiv sends \bot to the quiver with one object and no arrows, and nn to the quiver [n]=(01n)[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

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

Last revised on June 15, 2021 at 19:44:31. See the history of this page for a list of all contributions to it.