# Lax $\mathcal{F}$-adjunctions

## Idea

The notion of adjunction or 2-adjunction can be “laxified” in many ways; as discussed at lax 2-adjunction, one can make the triangle identities hold only up to a noninvertible cell, make the unit and counit only lax natural, and even consider making the functors only lax. Of these, lax naturality of the unit and counit is problematic because lax natural transformations do not satisfy a Yoneda lemma. However, this becomes less of a problem if the lax transformations restrict to pseudo ones on certain well-behaved subcategories. An abstract context to define this is that of F-categories.

## Definition

Let $K,L$ be F-categories (strict for simplicity). A lax $\mathcal{F}$-adjunction between them consists of:

• $\mathcal{F}$-functors $F:K\to L$ and $G:L\to K$ (strict for simplicity).
• pseudo/lax F-natural transformations$\eta : Id_K \to G F$ and $\epsilon : F G \to Id_L$.
• The triangle identities for $\eta,\epsilon$ hold up to isomorphism (which can be made coherent).

If $\epsilon$ is fully pseudo, we call it a right semi-lax $\mathcal{F}$-adjunction. Dually, we have left semi-lax, right semi-oplax, left semi-oplax, and so on. (We could also consider allowing the triangle identities or functors to be lax, but we will not.)

## Properties

The following theorem seems to fail for 2-adjunctions involving lax transformations; the fact that it holds for lax $\mathcal{F}$-adjunctions thus means that they are significantly better-behaved and more interesting. It was first observed (without the terminology of $\mathcal{F}$-categories) by Johnstone.

###### Theorem

If $F:K\to L$ is an $\mathcal{F}$-functor and $G,G'$ are two lax right $\mathcal{F}$-adjoints for it, then $G\simeq G'$.

###### Proof

As usual, we consider the composites

$G \xrightarrow{\eta' G} G' F G \xrightarrow{G' \epsilon} G' \qquad G' \xrightarrow{\eta G'} G F G' \xrightarrow{G \epsilon'} G.$

The usual proof of uniqueness of adjoints applies to show that these are pseudo-inverses, but we do need the fact that these unit and counit are pseudo/lax rather than merely lax. For the proof requires using naturality squares to commute units and counits past each other, and since their components are tight and they are all pseudo-natural on tight morphisms these naturality squares commute up to isomorphism; if they only commuted laxly then the proof would fail.

Note that a priori these composites are themselves also only pseudo/lax $\mathcal{F}$-natural. However, any lax natural transformation that is (pseudo) invertible is actually pseudo natural, so in fact these composites are a pseudo/pseudo $\mathcal{F}-$natural equivalence $G\simeq G'$.

## Examples

### Fibrations in a 2-category

In Johnstone it is shown that a fibration in a 2-category $K$ can equivalently be defined as a morphism $p:E\to B$ with the following property. Let $K\swarrow B$ denote the oplax slice 2-category, whose objects are morphisms $f:A\to B$ and whose morphisms are pairs $(g:A\to A', \alpha : f' g \Rightarrow f)$. We make $K\swarrow B$ into an $\mathcal{F}$-category by declaring $(g,\alpha)$ to be tight if $\alpha$ is an isomorphism.

Now, composing with $p:E\to B$ induces an $\mathcal{F}$-functor $\Sigma_p : K\swarrow E \to K\swarrow B$.

###### Theorem

$p$ is a fibration (in the pseudo sense) if and only if $\Sigma_p$ has a right semi-lax $\mathcal{F}$-adjoint.