This page is about conjunctions in double (or higher) categories; see logical conjunction for the meet of truth values.
A conjunction (sometimes called adjunction, e.g. by Grandis and Parè) 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.
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
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 $g$.
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.
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.
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.
The concept is due to (there called orthogonal adjoint):
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
Last revised on August 7, 2024 at 08:40:58. See the history of this page for a list of all contributions to it.