Contents

category theory

## Applications

#### 2-Category theory

2-category theory

Definitions

Transfors between 2-categories

Morphisms in 2-categories

Structures in 2-categories

Limits in 2-categories

Structures on 2-categories

# Contents

## Definition

An adjoint triple $F \dashv G \dashv H$ is called an ambidextrous adjunction (or sometimes ambiadjunction or ambijunction, for short) if the left adjoint $F$ and the right adjoint $H$ of $G$ are equivalent $F \simeq H$, or more precisely: equipped with a specified equivalence.

In fact, often $F$ is identified with $H$, which is the situation of astrongly adjoint pair* $(F \dashv G \dashv F)$ originally considered by Morita 1965. Some authors refer to this situation by saying that $G$ is a Frobenius functor (ie. a functor which has a left adjoint that is also a right adjoint).

Sometimes $F$ is said to be biadjoint to $G$ (not to be confused with biadjoint in the sense of biadjunction). Functor $G$ which has a left and right adjoint which are equivalent is said to be Frobenius functor.

In the special case that $G$ is a fully faithful functor with an ambidextrous adjoint one also speaks of an essential localization (cf. bireflective subcategory).

## Properties

### Frobenius algebra structure

The monad induced by an ambidextrous adjunction is a Frobenius monoid object in endofunctors. (e.g. Lauda 05, theorem 17), hence a Frobenius monad.

### Fiberwise characterization of ambidextrous Kan extension

Let $\mathcal{D} \in Cat_\infty$ be an (∞,1)-category with small (∞,1)-colimits. For $f \;\colon\; X \longrightarrow Y$ a morphism of ∞-groupoids, write

$f^\ast \;\colon\; [Y,\mathcal{D}] \longrightarrow [X,\mathcal{D}]$

for the induced pullback of (∞,1)-functor (∞,1)-categories (which one may think of as the categories of $\mathcal{D}$-valued local systems over $X$ and $Y$, respectively). The left adjoint and right adjoint (if it exists) of this are left and right (∞,1)-Kan extension.

###### Definition

Say that a morphism $f$ is $\mathcal{D}$-ambidextrous if $(f_! \dashv f^\ast)$ is an ambidextrous adjunction $(f_! \simeq f_\ast)$ and in addtion all pullbacks of $f$ satisfy some property (…).

Say that an ∞-groupoid $A \in Grpd_\infty$ is $\mathcal{D}$-ambidextrous if its terminal map is.

###### Proposition

A morphism $f \colon X \to Y$ between ∞-groupoids, is $\mathcal{D}$-ambidextrous, def. , precisely if each homotopy fiber $X_y$ of $f$ is.

## Examples

###### Example

(coincident limits and colimits)

Let $\mathcal{C}$ be a small category and $\mathcal{D}$ any category and consider the functor $const \mathcal{D} \longrightarrow [\mathcal{C}^{op}, \mathcal{D}]$ that sends objects to constant presheaves with this value. Then the right adjoint of this functor is, if it exists, the limit construction, and the left adjoint is, if it exists, the colimit construction. (See also at Kan extension.) Therefore if both exist as an ambidextrous adjunction, then this means that limits in $\mathcal{D}$ over diagrams of shape $\mathcal{C}$ coincide with the colimits over these diagrams. If $\mathcal{C}$ is a finite set, then this situation is traditionally referred to as biproducts. Generally therefore this is sometimes called bilimits (but see the discussion of the terminology there).

In (Hopkins-Lurie 14, section 4.3) such $\mathcal{C}$ is called $\mathcal{D}$-ambidextrous (or rather, they consider $\mathcal{C}$ an ∞-groupoid and hence call it a $\mathcal{D}$-ambidextrous space). Concrete examples of this include those discussed at K(n)-local stable homotopy theory.

###### Example

(Yoga of six functors)

A Wirthmüller context in the presence of an un-twisted Wirthmüller isomorphism is an ambidextrous adjunction.

###### Example

Ambidextrous adjunctions were maybe first considered under the name strongly adjoint pairs (of functors), in:

• Kiiti Morita, Adjoint pairs of functors and Frobenius extensions, Science Reports of the Tokyo Kyoiku Daigaku, Section A 9 202/208 (1965) 40-71 [jstor:43698658]

The terminology “Frobenius functors” for “strongly adjoint pairs” is due to

The case of bireflective subcategories:

On the issue of equipping an ambidextrous adjunction $F \dashv G \dashv H$ with a specific equivalence between $F$ and $H$: