nLab
conjunction

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

---

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

$$\begin{array}{ccc}A& \stackrel{\mathrm{id}}{\to }& A\\ {}^f\downarrow & {}^{\eta }⇙& {\downarrow }^{\mathrm{id}}\\ B& \underset{g}{\to }& A\end{array}\mathrm{and}\begin{array}{ccc}B& \stackrel{g}{\to }& A\\ {}^{\mathrm{id}}\downarrow & {}^{\epsilon }⇙& {\downarrow }^f\\ B& \underset{\mathrm{id}}{\to }& B\end{array}$$\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 $\epsilon {\circ }_h\eta ={\mathrm{id}}_g$ and $\eta {\circ }_v\epsilon ={\mathrm{id}}_f$, 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 $\mathrm{Sq}(K)$ of squares (“quintets”) in any 2-category $K$, a conjunction is simply an internal adjunction in $K$.

• In the double category $T$-Alg of algebras, lax morphisms, and colax morphisms for a 2-monad, 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 $V{D}^{\mathrm{op}}\to HD$ 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