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



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.


Let f:ABf\colon A\to B be a vertical arrow and g:BAg\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

A id A f η id B g AandB g A id ε f B id B \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 ε hη=id g\varepsilon \circ_h \eta = id_g and η vε=id f\eta \circ_v \varepsilon = id_{f}, where h\circ_h and v\circ_v denote horizontal and vertical composition of 2-cells.

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


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

  • In the double category TT-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.


  • 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 DD has a right conjoint, then the functor fgf\mapsto g is a pseudofunctor VD opHDV 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 DD into a proarrow equipment, or equivalently a framed bicategory.


Last revised on February 27, 2018 at 09:03:16. See the history of this page for a list of all contributions to it.