nLab multi-adjoint




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.

Every multiadjunction induces a multimonad.



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


Of course, if each set I(x)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.


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


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

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


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

Rχα(i,ϕ)=RχRϕη x,i=α(i,χϕ). R\chi \circ \alpha(i, \phi) = R\chi \circ R\phi \circ \eta_{x,i} = \alpha(i, \chi \circ \phi).

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

For the following theorem, let Fam(D)Fam(D) be the category

  • whose objects are indexed sets of objects of DD, hence pairs (I,h)(I, h) where II is a set and h:IObj(D)h \colon I \to Obj(D),

  • whose morphisms (f,ϕ):(I,h)(I,h)(f, \phi) \colon (I, h) \to (I', h') consist of a function f:IIf \colon I' \to I (note the contravariance) and ϕ:(iI)Hom D(h(f(i)),h(i))\phi: (i \in I') \to Hom_D\Big(h\big(f(i)\big), h'(i)\Big).

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

Hence Fam(D)Fam(D) is the free cartesian monoidal category over DD, with JJ the unit of the free cartesian monoidal category monad.

(This is dual to the free coproduct completion, an example of a Grothendieck construction, see here.)


A multi-adjoint to RcolDCR \col D \to C is a functor K:CFam(D)K : C \to Fam(D) such that RR is J J -coadjoint to K K , meaning that

α:Hom Fam(D)(Kc,Jd)Hom C(c,Rd), \alpha : Hom_{Fam(D)}(K c, J d) \cong Hom_{C}(c, R d),

naturally in cc and dd.


We use the characterization in theorem .

First of all, note that the type of the object part of KK is the same as the type of (I,L)(I, L); let us identify these. Next, note that Hom Fam(D)(Kc,Jd)Σ(iI(x)).Hom D(L(c,i),d)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 KK and naturality in cc.

Conversely, assume that the object part of KK is given. Then we construct a morphism part such that α\alpha is natural in cc.

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

(iI(c))Σ(iI(c)).Hom D(L(c,i),L(c,i)), (i' \in I(c')) \to \Sigma(i \in I(c)).Hom_D(L(c, i), L(c', i')),


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

so we can take K(ϕ,i)=α 1(η c,iϕ)K(\phi, i') = \alpha^{-1}(\eta_{c', i'} \circ \phi). Denote the components of KϕK\phi as Iϕ:I(c)I(c)I\phi : I(c') \to I(c) and Lϕ:(iI(i))Hom D(L(c,I(ϕ,i)),L(c,i))L\phi : (i' \in I(i')) \to Hom_D(L(c, I(\phi, i')), L(c', i')).

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

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

    KχKϕ=(IϕIχ,λi.L(χ,i)L(ϕ,I(χ,i)))K \chi \circ K \phi = (I\phi \circ I\chi, \lambda i''.L(\chi, i'') \circ L(\phi, I(\chi, i'')))

    =λi.(I(ϕ,I(χ,i)),L(χ,i)L(ϕ,I(χ,i)))= \lambda i''.(I(\phi, I(\chi, i'')), L(\chi, i'') \circ L(\phi, I(\chi, i'')))

    =λi.JL(χ,i)(I(ϕ,I(χ,i)),L(ϕ,I(χ,i)))= \lambda i''.JL(\chi, i'') \circ (I(\phi, I(\chi, i'')), L(\phi, I(\chi, i'')))

    =λi.JL(χ,i)K(ϕ,I(χ,i))= \lambda i''.JL(\chi, i'') \circ K(\phi, I(\chi, i''))

    =λi.JL(χ,i)α 1(η c,I(χ,i)ϕ)= \lambda i''.JL(\chi, i'') \circ \alpha^{-1}(\eta_{c', I(\chi, i'')} \circ \phi)

    =λi.α 1(RL(χ,i)α(I(χ,i),id c)ϕ)= \lambda i''.\alpha^{-1}(RL(\chi, i'') \circ \alpha(I(\chi, i''), id_{c'}) \circ \phi)

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

    =λi.α 1(α(K(χ,i))ϕ)= \lambda i''.\alpha^{-1}(\alpha(K(\chi, i'')) \circ \phi)

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

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

α(ψKϕ)=α((i,χ)Kϕ)=α(I(ϕ,i),χL(ϕ,i))\alpha(\psi \circ K\phi) = \alpha((i', \chi) \circ K\phi) = \alpha(I(\phi, i'), \chi \circ L(\phi, i'))

=α(JχK(ϕ,i))=α(Jχα 1(η c,iϕ))= \alpha(J\chi \circ K(\phi, i')) = \alpha(J\chi \circ \alpha^{-1}(\eta_{c', i'} \circ \phi))

=Rχη c,iϕ=α(Jχ(i,id c))ϕ=α(i,χ)ϕ=α(ψ)ϕ= 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.


A functor has a left multiadjoint if and only if it factors (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 RR factors into a final functor followed by a discrete fibration via the comprehensive factorisation. If RR has a left multiadjoint, the final functor will in addition be right adjoint.

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



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

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

Conversely, if DD has a terminal object and R:DCR \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)Σ(iI(c)).Hom D(L(c,i),)I(c). Hom_C(c, R\top) \cong \Sigma(i \in I(c)).Hom_D(L(c, i), \top) \cong I(c).

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


The category Field of fields has a multi-initial object, i.e. the functor Fld1Fld \to 1 has a left multi-adjoint (Def. ). This consists of all the “prime fields”, \mathbb{Q} and 𝔽 p\mathbb{F}_p for some prime number pp.


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) [Categories localisables.pdf]

  • 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).

See also:

Last revised on May 22, 2024 at 12:32:07. See the history of this page for a list of all contributions to it.