nLab
ambidextrous adjunction

Contents

Contents

Definition

An adjoint triple F⊣G⊣HF \dashv G \dashv H is called an ambidextrous adjunction (or sometimes ambijunction, for short) if the left adjoint FF and the right adjoint HH of GG are equivalent F≃HF \simeq H.

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 ambid. Kan extension

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

f *:[Y,π’ž]⟢[X,π’ž] f^\ast \;\colon\; [Y,\mathcal{C}] \longrightarrow [X,\mathcal{C}]

for the induced pullback of (∞,1)-functor (∞,1)-categories (which one may think of as the categories of π’ž\mathcal{C}-valued local systems over XX and YY, respectively). The left adjoint and right adjoint (if it exists) of this are left and right (∞,1)-Kan extension.

Definition

Say that a morphism ff is π’Ÿ\mathcal{D}-ambidextrous if (f !⊣f *)(f_! \dashv f^\ast) is an ambidextrous adjunction (f !≃f *)(f_! \simeq f_\ast) and in addtion all pullbacks of ff satisfy some property (…).

Say that an ∞-groupoid A∈Grpd ∞A \in Grpd_\infty is π’Ÿ\mathcal{D}-ambidextrous if its terminal map is.

(Hopkins-Lurie 14, def. 4.1.11)

Proposition

A morphism f:Xβ†’Yf \colon X \to Y between ∞-groupoids, is π’Ÿ\mathcal{D}-ambidextrous, def. , precisely if each homotopy fiber X yX_y of ff is.

(Hopkins-Lurie 14, prop. 4.3.5)

Examples

Example

(coincident limits and colimits)

Let π’ž\mathcal{C} be a small category and π’Ÿ\mathcal{D} any category and consider the functor constπ’ŸβŸΆ[π’ž op,π’Ÿ]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 adjounction, then this means that limits in π’Ÿ\mathcal{D} over diagrams of shape π’ž\mathcal{C} coincide with the colimits over these diagrams. If π’ž\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 π’ž\mathcal{C} is called π’Ÿ\mathcal{D}-ambidextrous (or rather, they consider π’ž\mathcal{C} an ∞-groupoid and hence call it a π’Ÿ\mathcal{D}-ambidextrous space). Concrete examples of this include those discussed at K(n)-local stable homotopy theory.

Example

(Yoga of six functors)

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

References

Last revised on January 31, 2014 at 10:43:28. See the history of this page for a list of all contributions to it.