nLab
Kan lift

Contents

Idea

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).

Informally, a Kan lift is a best approximate solution to the problem of finding a lift f˜:AB\widetilde{f}: A \to B of an arrow (morphism) f:ACf: A \to C through an arrow p:BCp: B \to C, as in the diagram

B p A f C\array{ & & B \\ & & \downarrow p \\ A & \overset{f}{\to} & C }

Of course, lifts typically don’t literally exist in the sense of an equation pf˜=fp \circ \widetilde{f} = f or an isomorphism pf˜fp \circ \widetilde{f} \cong f. But in good situations, one may have the next best thing: a 2-cell pf˜fp \circ \widetilde{f} \Rightarrow f 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.

terminology

Usually, when working within a 2-categorical context, Kan lifts are simply refered to as lifts/liftings, just as what happens with Kan extensions.

Definition

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.

Local description

Given 1-cells p:BCp: B \to C, f:ACf: A \to C in a bicategory, a right Kan lift of ff through pp, denoted Rift pfRift_p f, is a 1-cell f˜:AB\widetilde{f}: A \to B equipped with a 2-cell

ε:pf˜f\varepsilon: p \circ \widetilde{f} \Rightarrow f

satisfying the universal property: given any pair (g:AB,η:pgf)(g: A \to B, \eta: p \circ g \Rightarrow f), there exists a unique 2-cell

ζ:gf˜\zeta: g \Rightarrow \widetilde{f}

such that ε(pζ)=η\varepsilon \bullet (p \circ \zeta) = \eta. (Here \circ refers to composition across a 0-cell, and \bullet to composition across a 1-cell.) As with any universal description, the pair (Rift pf,ε)(Rift_p f, \varepsilon) is unique up to unique 2-cell isomorphism.

A left Kan lift of ff through pp, denoted Lift pfLift_p f, is a 1-cell f˜:AB\widetilde{f}: A \to B equipped with a 2-cell

η:fpf˜\eta: f \Rightarrow p \circ \widetilde{f}

such that given any pair (g:AB,θ:fpg)(g: A \to B, \theta: f \Rightarrow p \circ g), there exists a unique 2-cell

ζ:f˜g\zeta: \widetilde{f} \Rightarrow g

such that (pζ)η=θ(p \circ \zeta) \bullet \eta = \theta.

Global description

Given p:BCp: B \to C and a 0-cell AA in a bicategory, if the right Kan lift Rift pfRift_p f exists for any f:ACf: A \to C, then we speak of a global Kan lift. When this is the case, we may define a functor between hom-categories

Rift p:[A,C][A,B]Rift_p: [A, C] \to [A, B]

which at the object level is of course fRift pff \mapsto Rift_p f. At the morphism level, given a 2-cell

η:fg:AC\eta: f \Rightarrow g: A \to C

there is an induced 2-cell Rift pfRift pgRift_p f \Rightarrow Rift_p g, the one which corresponds (by the universal property of Rift pgRift_p g) to the composite

pRift pfεfηg.p \circ Rift_p f \overset{\varepsilon}{\Rightarrow} f \overset{\eta}{\Rightarrow} g.

A standard universality argument shows that Rift pRift_p thus defined is functorial.

Proposition: Rift pRift_p is right adjoint to the functor [A,p]:[A,B][A,C][A, p]: [A, B] \to [A, C] obtained by postcomposing with pp, i.e., the functor fpff \mapsto p \circ f.

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.

  • In a post at the nCafé, David Corfield put the question: why are Kan extensions more frequently mentioned than Kan lifts? One proposed answer is that in many cases a left or right Kan lift will exist for a trivial reason: namely that pp itself has a left or right adjoint. Other examples will be given below.

Properties

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:

  • given (f˜,ε)=Rift pf(\widetilde{f}, \varepsilon) = \mathop{Rift}_p f, gg is said to respect this right lift if (f˜g,εg)=Rift p(fg)(\widetilde{f}g, \varepsilon \bullet g) = \mathop{Rift}_p(fg)

and analogously for left Kan lifts. A Kan lift f^\widehat{f} is absolute if it is respected by any 1-cell into dom(f^)\mathop{dom}(\widehat{f}). 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

  • Let ModMod denote the framed bicategory of small categories CC, bimodules r:CDr: C \to D (i.e., functors r:D op×CSetr: D^{op} \times C \to Set), and transformations between such bimodules. Then global right Kan lifts exist (as do global right Kan extensions). These may be computed as weighted limits or via end formulas, e.g.,
    (Rift rs)(b,c)= d:Dhom(r(d,c),s(d,b))(Rift_r s)(b, c) = \int_{d: D} hom(r(d, c), s(d, b))

Examples of this construction abound in mathematics, especially when generalized to the enriched category theory context. For example, in the bicategory RelRel, which corresponds to enrichment in 2\mathbf{2}, the right Kan lift is essentially a universally quantified predicate of the form

d:Dr(d,c)s(d,b)\forall_{d: D} r(d, c) \Rightarrow s(d, b)

(“for all dd satisfying condition rr, we impose condition ss”).

  • More generally, a biclosed bicategory? is precisely a bicategory where global right Kan extensions and right Kan lifts exist for every 1-cell pp. Monoidal bicategories provide instances of this.

  • adjunctions in a 2-category can be defined in terms of Kan lifts: a 1-cell u:ABu\colon A \to B has a left adjoint iff Lift u1 B\mathop{Lift}_u 1_B exists and is absolute; in this case putting (f,ι)=Lift u1 B(f,\iota) = \mathop{Lift}_u 1_B we have fuf \dashv u with unit ι:1 Buf\iota \colon 1_B \Rightarrow u f. 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 Cat\mathbf{Cat} setting.

  • representably fully faithful 1-cells, meaning those for which B(X,f)B(X,f) is fully faithful in Cat\mathbf{Cat} for every object X:BX\colon B, are those for which (1 A,1 f)=Lift ff(1_A, 1_f) = \mathop{Lift}_f f, and this lifting is absolute.

  • In CatCat, if AA is small and BB is locally small, and if F:ABF: A \to B is a functor, then we have a Yoneda embedding y:APA=Set A opy: A \to P A = Set^{A^{op}} and a functor B(F,):BPAB(F-, -): B \to P A, and there is a canonical map

    y AB(F,)fy_A \to B(F-, -) \circ f

    (essentially, hom(a,b)hom(Fa,Fb)hom(a, b) \to hom(F a, F b) taking f:abf: a \to b to Ff:FaFbF f: F a \to F b). This arrow exhibits FF as a left Kan lift of yy through B(F,)B(F-, -), 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.

Revised on March 7, 2012 08:51:40 by Eduardo Pareja-Tobes? (83.44.177.12)