The notion of Kan lift in a 2-category or in a bicategory is dual to the notion of Kan extension, by a process of turning the 1-cells around, much in the same way as a homotopy lifting property is dual to a homotopy extension property (in model category theory, for instance).
Of course, lifts typically don’t literally exist in the sense of an equation or an isomorphism . But in good situations, one may have the next best thing: a 2-cell which is universal among 2-cells of this form. This gives the notion of right Kan lift. The notion of left Kan lift is similar, but with 2-cells in the opposite direction.
Usually, when working within a 2-categorical context, Kan lifts are simply refered to as lifts/liftings, just as what happens with Kan extensions.
As in the discussion at Kan extension, there is a local notion of Kan lift and a global notion. Expanding on the informal description above, we give the local notion first, followed by the (conceptually more perspicuous) global notion in terms of adjoint functors.
Given 1-cells , in a bicategory, a right Kan lift of through , denoted , is a 1-cell equipped with a 2-cell
satisfying the universal property: given any pair , there exists a unique 2-cell
such that . (Here refers to composition across a 0-cell, and to composition across a 1-cell.) As with any universal description, the pair is unique up to unique 2-cell isomorphism.
A left Kan lift of through , denoted , is a 1-cell equipped with a 2-cell
such that given any pair , there exists a unique 2-cell
such that .
Given and a 0-cell in a bicategory, if the right Kan lift exists for any , then we speak of a global Kan lift. When this is the case, we may define a functor between hom-categories
which at the object level is of course . At the morphism level, given a 2-cell
there is an induced 2-cell , the one which corresponds (by the universal property of ) to the composite
A standard universality argument shows that thus defined is functorial.
Proposition: is right adjoint to the functor obtained by postcomposing with , i.e., the functor .
This is the easier, more conceptual way to remember right Kan lifts: in a global sense, they are right adjoint to postcomposition. Similarly, global left Kan lifts are left adjoint to postcomposition.
Dual to the situation in Kan extensions, one is interested in whether a Kan lift is respected by a 1-cell with codomain the domain of the lift. This is defined as follows:
and analogously for left Kan lifts. A Kan lift is absolute if it is respected by any 1-cell into . Absolute Kan lifts subsume adjunctions and relative adjunctions, and are prominently present in the axioms of a Yoneda structure?; for more see the examples below.
Examples of this construction abound in mathematics, especially when generalized to the enriched category theory context. For example, in the bicategory , which corresponds to enrichment in , the right Kan lift is essentially a universally quantified predicate of the form
(“for all satisfying condition , we impose condition ”).
More generally, a biclosed bicategory? is precisely a bicategory where global right Kan extensions and right Kan lifts exist for every 1-cell . Monoidal bicategories provide instances of this.
adjunctions in a 2-category can be defined in terms of Kan lifts: a 1-cell has a left adjoint iff exists and is absolute; in this case putting we have with unit . The universal property of the left Kan lift plus absoluteness are enough to construct the counit and to verify the triangular equations. There’s of course a dual definition in terms of absolute right Kan lifts.
relative adjoints can also be expressed as absolute kan lifts; in fact, they are the same thing. See relative adjoint for a precise statement in the setting.
representably fully faithful 1-cells, meaning those for which is fully faithful in for every object , are those for which , and this lifting is absolute.
In , if is small and is locally small, and if is a functor, then we have a Yoneda embedding and a functor , and there is a canonical map
(essentially, taking to ). This arrow exhibits as a left Kan lift of through , which is moreover absolute. This example is important in the theory of Yoneda structures, due to Street and Walters; see Mark Weber’s updated development in the context of 2-topos theory.