nLab multi-adjoint

Idea

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

Definition

Definition

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

Remark

Of course, if each set $I$ is 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.

Examples

Example

Every parametric right adjoint with locally small codomain has a left multi-adjoint. And conversely, if $D$ has a terminal object and $G \colon D\to C$ has a left multi-adjoint, then it is a parametric right adjoint.

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.

Multi-monads

Any functor $U: A\to B$ which has a left multi-adjoint generates a multi-monad on $B$. Categories $A$ which can be reconstructed from this multi-monad are called multi-monadic (Diers 80).

Multi-monadic categories on $Set$ can be characterized in the following way: they are regular, with connected limits, with coequalizers of coequalizable pairs, their equivalence relations are effective, their forgetful functors preserve coequalizers of equivalence relations and reflect isomorphisms. Unlike monadic categories they need not have products. Examples include local rings, fields, inner spaces, locally compact spaces, locally compact groups, and complete ordered sets. (Diers 80, p.153)

Diers spectrum

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

References

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).

See also:

• Axel Osmond, On Diers theory of Spectrum I: Stable functors and right multi-adjoints, (arXiv:2012.00853)

• Axel Osmond, On Diers theory of Spectrum II: Geometries and dualities, (arXiv:2012.02167)

• Yves Diers, Multimonads and multimonadic categories, Journal of Pure and Applied Algebra 17 (1980) 153-170 (pdf)