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 be a vertical arrow and a horizontal arrow in a double category. These arrows are said to be a conjunction if they come equipped with 2-cells
such that and , where and denote horizontal and vertical composition of 2-cells.
Given such a conjunction, we say that and are conjoints of each other, and that is the right conjoint of and that is the left conjoint of .
In the double category of squares (quintets) in any 2-category , a conjunction is simply an internal adjunction in .
In the double category - 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 has a right conjoint, then the functor is a pseudofunctor 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 into a proarrow equipment, or equivalently a framed bicategory.
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 March 6, 2024 at 11:33:32. See the history of this page for a list of all contributions to it.