Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
A companion pair in a double category is a way of saying that a loose arrow and a tight morphism are “isomorphic”, even though they do not live in the same 1-category/2-category.
A connection pair in a double category is a strictly 2-functorial choice of companion pairs for every tight morphism.
Let be a tight morphism and a loose arrow in a double category. These are said to be a companion pair if they come equipped with 2-morphisms of the form: such that and , where and denote, respectively, loose and tight composition of 2-cells.
Given such a companion pair, we say that and are companions of each other. A double category for which every tight morphism admits a companion may be called companionable.
In the double category of squares (quintets) in any 2-category , a companion pair is simply an invertible 2-cell between two parallel 1-morphisms of .
In the double category - of algebras, lax morphisms, and colax morphisms for a 2-monad , an arrow (of either sort) has a companion precisely when it is a strong (= pseudo) -morphism. This is important in the theory of doctrinal adjunction.
The loose (or tight) dual of a companion pair is a conjunction.
Companion pairs (and conjunctions) have a mate correspondence generalizing the calculus of mates in 2-categories.
If every tight arrow in some double category has a companion, then the functor is a pseudofunctor from the tight 2-category to the loose one, which is the identity on objects, and locally fully faithful by the mate correspondence. A choice of companions that make this a strict 2-functor is called a connection on (an arbitrary choice of companions may be called a “pseudo-connection”). A double category with a connection is thereby equivalent to an F-category. If every tight arrow also has a conjoint, then this makes into a proarrow equipment, or equivalently a framed bicategory.
Companion pairs and mate-pairs of 2-cells between them in any double category form a 2-category . The functor is right adjoint to the functor sending a 2-category to its double category of squares.
A double category , carried by the span of categories has all companions iff such span is a two-sided fibration in the sense of Street. Indeed, consider a tight map in , thus a morphism in , and the unit loose arrow of : since is an opfibration, we obtain a square , with source (top boundary) and target (bottom boundary) necessarily since -cartesian maps are -vertical in a two-sided fibration. Dually, we get a cartesian square . We claim and are, respectively, the unit and counit of a companionship between and . First, observe this latter equation holds since the identiy square of factors as , using either universal property. This proves also the first companionship equation. As for the fact , it follows again from the aforementioned factorization: is an -vertical morphism which is thus isomorphic to the identity of by uniqueness of the -vertical, -vertical, -vertical factorization of a morphism in . Vice versa, companions can be used to construct the above co/cartesian lifts, as shown in (Shulman ‘08).
Marco Grandis and Robert Pare, Adjoints for double categories, NUMDAM
Robert Dawson and Robert Pare and Dorette Pronk, The Span construction, TAC.
Michael Shulman, Framed bicategories and monoidal fibrations, TAC
This latter reference explains the relationship between companions to connection pairs and foldings:
Ronnie Brown and C.B. Spencer, Double groupoids and crossed modules, Cahiers de Topologie et Géométrie Différentielle Catégoriques 17 (1976), 343–362.
Ronald Brown and Ghafar H. Mosa, Double categories, 2-categories, thin structures and connections, Theory and Application of Categories 5.7 (1999): 163-1757.
Thomas M. Fiore, Pseudo Algebras and Pseudo Double Categories, Journal of Homotopy and Related Structures, Volume 2, Number 2, pages 119-170, 2007. 51 pages.
Robert Pare, Seeing double, Talk given at FMCS 2018, (pdf)
Last revised on October 30, 2024 at 11:55:25. See the history of this page for a list of all contributions to it.