nLab conjunction

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

---

Idea

A conjunction (sometimes called adjunction, e.g. by Grandis and Parè) in a double category is a way of saying that a tight and a loose arrow are adjoint, even though they do not live in the same 2-category.

Definition

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

such that $\varepsilon \circ_\lambda \eta = id_f$ and $\eta \circ_\tau \varepsilon = id_g$, where $\circ_\lambda$ and $\circ_\tau$ denote, respectively, loose and tight 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 $L D^{op}\to T D$ from the loose bicategory to the tight 2-category, which is the identity on objects, and locally fully faithful by the mate correspondence. If every tight arrow also has a companion, then this makes $D$ into a proarrow equipment, or equivalently a framed bicategory.

• A double category $\mathbb{C}$ admits all conjoints iff the underlying spans $C_0 \xleftarrow{s} C_1 \xrightarrow{t} C_0$ is a two-sided opfibration, i.e. if its opposite span is a two-sided fibration. See at companion pair for a sketch of the proof of why this is the case.

Related concepts

• adjunction
• companion pair

References

The concept is due to (there called orthogonal adjoint):

• Marco Grandis and Robert Pare, Adjoints for double categories

The terminology “conjoint” is due to:

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

• Michael Shulman, Framed bicategories and monoidal fibrations, TAC