### Parametric adjoints = multi-adjoints

#### Posted by Mike Shulman

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

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.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.

## Re: Parametric adjoints = multi-adjoints

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 localisablescontains a number of useful results in a similar vein, but I haven’t been able to find a copy.)