Parametric adjoints = multi-adjoints
Posted by Mike Shulman
Here are two “weakenings” of the notion of adjoint functor:
If has a terminal object , a functor is a parametric right adjoint if the induced functor is a right adjoint.
A functor has a left multi-adjoint if for each there is a (small) family of morphisms such that for any morphism , there is a unique pair of an index and a morphism such that .
I just noticed that these two definitions are almost exactly the same! Specifically, if has a terminal object and is locally small, then a functor is a parametric right adjoint if and only if it has a left multi-adjoint.
First suppose is a parametric right adjoint, with left adjoint of . Then given , let (which is a set since is locally small). Then each is a morphism , i.e. an object of , so we have and the unit a morphism in . Now suppose given . Then the composite is an element , making into a morphism of . Thus, it corresponds to a unique morphism , giving the unique factorization in the definition of multi-adjoint.
Conversely, suppose has a left multi-adjoint, and let be an object of . Let be given as in the definition of multi-adjoint; then there is a unique and a factorization of through (necessarily by the unique morphism ). Define . Then for any , we know that any map factors uniquely through for some . But it follows that the composite must factor through the same ; so if is a morphism in then we must have , so that factors uniquely through . Thus, is a left adjoint to . 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 factors as where is “-generic”. The definition of -generic involves a sort of restricted lifting property, but it’s easy to see that the units in a multi-adjoint are -generic, and that if -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 localisables contains a number of useful results in a similar vein, but I haven’t been able to find a copy.)