## March 8, 2021

#### Posted by Mike Shulman

Here are two “weakenings” of the notion of adjoint functor:

1. If $C$ has a terminal object $1$, a functor $F:C\to D$ is a parametric right adjoint if the induced functor $F_1 : C \to D/F 1$ is a right adjoint.

2. A functor $F:C\to D$ has a left multi-adjoint if for each $y\in D$ there is a (small) family of morphisms $\{ \eta_{y,i}:y \to F x_i\}_{i\in I}$ such that for any morphism $g:y\to F z$, there is a unique pair of an index $i\in I$ and a morphism $h:x_i \to z$ such that $g = F h \circ \eta_{y,i}$.

I just noticed that these two definitions are almost exactly the same! Specifically, if $C$ has a terminal object and $D$ is locally small, then a functor $F:C\to D$ is a parametric right adjoint if and only if it has a left multi-adjoint.

First suppose $F$ is a parametric right adjoint, with left adjoint $L : D/F1\to C$ of $F_1$. Then given $y\in D$, let $I = D(y, F1)$ (which is a set since $D$ is locally small). Then each $i\in I$ is a morphism $i:y\to F1$, i.e. an object of $D/F1$, so we have $L i \in C$ and the unit $\eta_{y,i}: y \to F L i$ a morphism in $D/F1$. Now suppose given $g:y\to F z$. Then the composite $y \to F z \to F 1$ is an element $i\in I$, making $g$ into a morphism of $D/F1$. Thus, it corresponds to a unique morphism $L i \to z$, giving the unique factorization in the definition of multi-adjoint.

Conversely, suppose $F$ has a left multi-adjoint, and let $k:y\to F 1$ be an object of $D/F1$. Let $\{ \eta_{y,i}:y \to F x_i\}_{i\in I}$ be given as in the definition of multi-adjoint; then there is a unique $i\in I$ and a factorization of $k$ through $\eta_{y,i}$ (necessarily by the unique morphism $x_i \to 1$). Define $L k = x_i$. Then for any $z\in C$, we know that any map $g:y\to F z$ factors uniquely through $\eta_{y,j}$ for some $j\in I$. But it follows that the composite $y \to F z \to F 1$ must factor through the same $\eta_{y,j}$; so if $g$ is a morphism in $D/F1$ then we must have $j=i$, so that $g$ factors uniquely through $L k$. Thus, $L$ is a left adjoint to $F_1$. This completes the proof.

In fact, this argument is essentially already in the literature. In Familial 2-functors and parametric right adjoints, Weber proved (Prop. 2.6) that a functor whose domain has a terminal object is a parametric right adjoint if and only if every morphism $f:y\to F z$ factors as $y \xrightarrow{g} F x \xrightarrow{F h} F z$ where $g$ is “$F$-generic”. The definition of $F$-generic involves a sort of restricted lifting property, but it’s easy to see that the units $\eta_{y,i}$ in a multi-adjoint are $F$-generic, and that if $F$-generic factorizations exist then a representative set of their isomorphism classes forms a multi-adjoint (up to size questions).

However, I haven’t seen this stated before using the terminology “multi-adjoint”. In particular, it’s interesting that multi-adjoints can be thought of as a generalization of parametric adjoints to categories lacking a terminal object.

Posted at March 8, 2021 4:40 PM UTC

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I haven’t seen this relationship stated so directly, though Charles Walker observes in Lax familial representability and lax generic factorizations that familial functors are precisely those functors admitting left multi-adjoints; that familial functors are equivalently those admitting generic factorisations is attributed in Proposition 2.8 to Yves Diers. It’s very helpful to make this explicit connection, though; it’s not at all obvious from either paper (or even from both in concert)!

(I suspect Diers’s thesis Catégories localisables contains a number of useful results in a similar vein, but I haven’t been able to find a copy.)

Posted by: Nathanael Arkor on March 9, 2021 2:08 AM | Permalink | Reply to this

I thought of Diers too. There’s a good bit about familial representability and parametric right adjoints (though not under that name) in this paper, which seems to me to be underappreciated:

Aurelio Carboni and Peter Johnstone. Connected limits, familial representability and Artin glueing. Mathematical Structures in Computer Science 5 (1995), 441–459.

Since I’m citing this, I should also cite the corrigenda, though they’re irrelevant for the present purposes:

Aurelio Carboni and Peter Johnstone. Corrigenda for “Connected limits, familial representability and Artin glueing”. Mathematical Structures in Computer Science 14 (2004), 185–187.

Posted by: Tom Leinster on March 9, 2021 5:23 PM | Permalink | Reply to this

Great, post! What’s the typical reference for multi-adjoints?

I think I learned about them from Adamek-Rosicky Locally presentable and accessible categories, where they (along with weak adjoints) are used to characterize certain classes of accessible categories that are more general than locally presentable categories (“locally multi-presentable categories”, etc.)

Posted by: Mike Shulman on March 9, 2021 3:17 PM | Permalink | Reply to this

Awesome, thanks!

Small question:

In $h : x_i \to y$ is $y$ supposed to be $z$?

Yep, thanks. I’ll fix it in the post.

Posted by: Mike Shulman on March 9, 2021 4:46 PM | Permalink | Reply to this

Anything of interest here?

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

Diers developed a general theory of right multiadjoint functors leading to a purely categorical, point-set construction of spectra. Situations of multiversal properties return sets of canonical solutions rather than a unique one. In the case of a right multiadjoint, each object deploys a canonical cone of local units jointly assuming the role of the unit of an adjunction. This first part revolves around the theory of multiadjoint and recalls or precises results that will be used later on for geometric purpose. We also study the weaker notion of local adjoint, proving Beck-Chevaley conditions relating local adjunctions and the equivalence with the notion of stable functor. We also recall the link with the free-product completion, and describe factorization aspects involved in a situation of multi-adjunction.

There’s a follow-up

• Axel Osmond, On Diers theory of Spectrum II: Geometries and dualities, (arXiv:2012.02167)
Posted by: David Corfield on March 10, 2021 7:48 AM | Permalink | Reply to this

Posted by: David Corfield on March 10, 2021 1:50 PM | Permalink | Reply to this

Naturally, multiadjoints have appeared on the Café before, in a comment here from Paul Taylor in the context of codensity monads.

Posted by: David Corfield on March 10, 2021 2:53 PM | Permalink | Reply to this

There is a paper by Walter Tholen on pro-categories and multi-adjoints that may be useful. (Can. J. Math., Vol. XXXVI, No. 1, 1984, pp. 144-155)

Posted by: Tim Porter on March 11, 2021 7:49 AM | Permalink | Reply to this

That could well contain Mike’s result. Theorem 2.4 claims that, in some circumstance, the condition of being a right $\mathcal{D}$-pro-adjoint and being a partial left adjoint relative to some embedding are equivalent.

When $\mathcal{D}$ is $Set$, the first condition is the same as being a right multiadjoint.

Posted by: David Corfield on March 11, 2021 12:02 PM | Permalink | Reply to this