category theory

## Idea

A multi-adjoint is like an adjoint functor but sends each object to a family of objects rather than a single one.

Under mild conditions it is equivalent to the notion of parametric right adjoint.

## Definition

###### Definition

Let $R \colon D\to C$ be a functor. We say $R$ has a left multi-adjoint if for each $x \in C$ there is a family of morphisms $\{\eta_{x,i} \colon x \to R L(x,i)\}_{i\in I(x)}$ such that for any morphism $f \colon x\to R y$ there exists a unique pair of an index $i\in I(x)$ and a morphism $g \colon L(x, i) \to y$ such that $f = R g \circ \eta_{x,i}$.

###### Remark

Of course, if each set $I(x)$ is always a singleton, then Def. reduces to the notion of an actual left adjoint. On the other hand, if we remove the uniqueness requirement, this reduces to the solution set condition. Thus, a multi-adjoint is “halfway between” the solution-set condition and the actual existence of an adjoint.

###### Remark

If in Def. we weaken the notion of adjunction in the opposite way, by removing the uniqueness requirement but keeping $I$ a singleton (or equivalently add to the solution-set condition that $I$ is a singleton), we obtain the notion of a weak adjoint.

###### Theorem

Having a left multi-adjoint can be equivalently characterized as follows:

$R : D \to C$ has a left multi-adjoint $(I, L)$ with $I : Obj(C) \to Set$ and $L : (x : Obj(C)) \to I(x) \to Obj(D)$ if for all $x$ and $y$, there is an isomorphism $\alpha : \Sigma(i \in I(x)).Hom_D(L(x, i), y) \cong Hom_C(x, Ry)$, natural in $y$.

###### Proof

We first prove the Hom-characterization from the definition. Given $(i, \phi : L(x, i) \to y)$, we get $\alpha(\phi) := R\phi \circ \eta_{x, i} : x \to Ry$. The definition demands that this is an isomorphism. To see naturality, let $\chi : y \to z$. Then

$R\chi \circ \alpha(i, \phi) = R\chi \circ R\phi \circ \eta_{x,i} = \alpha(i, \chi \circ \phi).$

Now let $\alpha$ be given. Then we define $\eta_{x, i} = \alpha(i, id) : x \to RL(x, i)$. By naturality, $\alpha(i, \phi) = R\phi \circ \eta_{x, i}$. Then invertibility of $\alpha$ proves the condition in the definition.

###### Theorem

Let $Fam(D)$ be the category

• whose objects are pairs $(I, h)$ such that $I$ is a set and $h : I \to Obj(D)$,

• whose morphisms $(f, \phi) : (I, h) \to (I', h')$ consist of a function $f : I' \to I$ (note the contravariance) and $\phi: (i \in I) \to Hom_D(h(i), h'(f(i)))$.

This is an example of a Grothendieck construction.

Define $J : D \to Fam(D) : d \mapsto (\{*\}, * \mapsto d)$.

$Fam(D)$ is also the free cartesian category over $D$, with $J$ the unit of the free cartesian category monad.

Then a multi-adjoint to $R : D \to C$ is a functor $K : C \to Fam(D)$ such that $R$ is the $J$-right adjoint to $K$, meaning that

$\alpha : Hom_{Fam(D)}(K c, J d) \cong Hom_{C}(c, R d),$

naturally in $c$ and $d$.

###### Proof

We use the characterization in theorem .

First of all, note that the type of the object part of $K$ is the same as the type of $(I, L)$; let us identify these. Next, note that $Hom_{Fam(D)}(K c, J d) \cong \Sigma(i \in I(x)).Hom_D(L(c, i), d)$.

So to show that the characterization here implies the definition, we simply need to forget functoriality of $K$ and naturality in $c$.

Conversely, assume that the object part of $K$ is given. Then we construct a morphism part such that $\alpha$ is natural in $c$.

Take $\phi : Hom_C(c, c')$. We want $K\phi : Hom_{Fam(D)}(Kc, Kc')$, i.e.

$(i' \in I(c')) \to \Sigma(i \in I(c)).Hom_D(L(c, i), L(c', i')),$

i.e.

$(i' \in I(c')) \to Hom_{Fam(D)}(Kc, JL(c', i'))$

so we can take $K(\phi, i') = \alpha^{-1}(\eta_{c', i'} \circ \phi)$. Denote the components of $K\phi$ as $I\phi : I(c') \to I(c)$ and $L\phi : (i' \in I(i')) \to Hom_D(L(c, I(\phi, i')), L(c', i'))$.

• This preserves the identity: $K(id) = \lambda i.\alpha^{-1}(\eta_{x, i} \circ id) = \lambda i.(i, id)$.

• This preserves composition. Let $\phi : Hom_C(c, c')$ and $\chi : Hom_C(c', c'')$. With some abuse of notation:

$K \chi \circ K \phi = (I\phi \circ I\chi, \lambda i''.L(\chi, i'') \circ L(\phi, I(\chi, i'')))$

$= \lambda i''.(I(\phi, I(\chi, i'')), L(\chi, i'') \circ L(\phi, I(\chi, i'')))$

$= \lambda i''.JL(\chi, i'') \circ (I(\phi, I(\chi, i'')), L(\phi, I(\chi, i'')))$

$= \lambda i''.JL(\chi, i'') \circ K(\phi, I(\chi, i''))$

$= \lambda i''.JL(\chi, i'') \circ \alpha^{-1}(\eta_{c', I(\chi, i'')} \circ \phi)$

$= \lambda i''.\alpha^{-1}(RL(\chi, i'') \circ \alpha(I(\chi, i''), id_{c'}) \circ \phi)$

$= \lambda i''.\alpha^{-1}(\alpha(I(\chi, i''), L(\chi, i'')) \circ \phi)$

$= \lambda i''.\alpha^{-1}(\alpha(K(\chi, i'')) \circ \phi)$

$= \lambda i''.\alpha^{-1}(\eta_{c'', i''} \circ \chi \circ \phi)$.

Now we prove that $\alpha$ is natural in $c$. Let $\psi : Hom_{Fam(D)}(Kc', Jd)$ and $\phi : Hom_{C}(c, c')$. Note that $\psi$ is essentially of the form $(i', \chi)$ for $i' \in I(c')$ and $\chi \in \Hom_D(L(c', i), d)$. With more abuse of notation:

$\alpha(\psi \circ K\phi) = \alpha((i', \chi) \circ K\phi) = \alpha(I(\phi, i'), \chi \circ L(\phi, i'))$

$= \alpha(J\chi \circ K(\phi, i')) = \alpha(J\chi \circ \alpha^{-1}(\eta_{c', i'} \circ \phi))$

$= R\chi \circ \eta_{c', i'} \circ \phi = \alpha(J\chi \circ (i', id_{c'})) \circ \phi = \alpha(i', \chi) \circ \phi = \alpha(\psi) \circ \phi$.

###### Proposition

A functor has a left multiadjoint if and only if it facts (essentially uniquely) as a right adjoint functor followed by a discrete fibration.

Since both right adjoint functors and discrete fibrations have left multiadjoints, one direction follows from the closure of functors having left multiadjoints under composition.

In the other direction, every functor $R$ factors into a final functor followed by a discrete fibration via the comprehensive factorisation. If $R$ has a left multiadjoint, the final functor will in addition be right adjoint.

For a complete proof, see (Diers 80, Proposition 1.1).

## Examples

###### Example

Every parametric right adjoint $R : D \to C$ with locally small codomain has a left multi-adjoint, where

• $I(c) = Hom_C(c, R\top)$,
• $L$ is the left adjoint to $R^{/\top} : D \to C/R\top : d \mapsto (Rd, R())$.

Conversely, if $D$ has a terminal object and $R \colon D\to C$ has a left multi-adjoint, then it is a parametric right adjoint. Indeed, using the characterization in theorem , we see immediately that

$Hom_C(c, R\top) \cong \Sigma(i \in I(c)).Hom_D(L(c, i), \top) \cong I(c).$

Moreover, by naturality in $d$, for any morphism $\phi : Hom_C(c, Rd)$, the first component of $\alpha^{-1}(\phi)$ is fully determined by $R() \circ \phi : Hom_C(c, R\top)$.

###### Example

The category of fields has a multi-initial object, i.e. the functor $Fld \to 1$ has a left multi-adjoint (Def. ). This consists of all the “prime fields”, $\mathbb{Q}$ and $\mathbb{F}_p$ for some prime number $p$.

###### Example

The forgetful functor from the category of commutative local rings and local homomorphisms to the category of sets has a left multi-adjoint.

## Diers spectrum

The notion of a multiadjoint functor is used to define the Diers spectrum. See there for more details.

The notion of multi-adjoints is credited by Osmond 20a, 20b to:

• Yves Diers, Catégories localisables, PhD thesis. Paris 6 et Centre universitaire de Valenciennes et du Hainaut Cambrésis (1977).

• Yves Diers, Catégories localement multiprésentables, Archiv der Mathematik 34.1 (1980), pp. 344–356.

• Yves Diers, Une construction universelle des spectres, topologies spectrales et faisceaux structuraux, Communication in Algebra Volume 12, Issue 17-18 (1984) (doi:10.1080/00927878408823101)

(in discussion of the notion of spectra in geometry).