nLab conjunction

This page is about conjunctions in double (or higher) categories; see logical conjunction for the meet of truth values.

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Idea

A conjunction in a double category is a way of saying that a horizontal arrow and a vertical arrow are adjoint, even though they do not live in the same 2-category.

Definition

Let $f\colon A\to B$ be a vertical arrow and $g\colon B\to A$ a horizontal arrow in a double category. These arrows are said to be a conjunction if they come equipped with 2-cells

$$\array{ A & \overset{id}{\to} & A \\ ^f\downarrow & ^{\eta}\swArrow & \downarrow^{id} \\ B & \underset{g}{\to} & A} \qquad and\qquad \array{ B & \overset{g}{\to} & A \\ ^{id} \downarrow & ^{\varepsilon}\swArrow & \downarrow^f \\ B & \underset{id}{\to} & B }$$

such that $\varepsilon \circ_h \eta = id_g$ and $\eta \circ_v \varepsilon = id_{f}$, where $\circ_h$ and $\circ_v$ denote horizontal and vertical composition of 2-cells.

Given such a conjunction, we say that $f$ and $g$ are conjoints of each other, and that $g$ is the right conjoint of $f$ and that $f$ is the left conjoint of $g$.

Examples

• In the double category $\mathbf{Sq}(K)$ of squares (quintets) in any 2-category $K$, a conjunction is simply an internal adjunction in $K$.

• In the double category $T$-$\mathbf{Alg}$ of algebras, lax morphisms, and colax morphisms for a 2-monad $T$, an conjunction is precisely a doctrinal adjunction between a colax morphism an a lax morphism.

Remarks

• The horizontal (or vertical) dual of a conjunction is a companion pair.

• Conjunctions (and companion pairs) have a mate correspondence generalizing the calculus of mates in 2-categories.

• If every vertical arrow in some double category $D$ has a right conjoint, then the functor $f\mapsto g$ is a pseudofunctor $V D^{op}\to H D$ from the vertical 2-category to the horizontal one, which is the identity on objects, and locally fully faithful by the mate correspondence. If every vertical arrow also has a companion, then this makes $D$ into a proarrow equipment, or equivalently a framed bicategory.

References

• Marco Grandis and Robert Pare, Adjoints for double categories

• Robert Dawson and Robert Pare and Dorette Pronk, The Span construction, TAC.

• Michael Shulman, Framed bicategories and monoidal fibrations, TAC