nLab
Kan lift

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 $\tilde{f}:A\to B$ of an arrow $f:A\to C$ through an arrow $p:B\to C$, as in the diagram

$$\begin{array}{ccc}& & B\\ & & \downarrow p\\ A& \stackrel{f}{\to }& C\end{array}$$\array{ & & B \\ & & \downarrow p \\ A & \overset{f}{\to} & C }

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

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 $p:B\to C$, $f:A\to C$ in a bicategory, a right Kan lift of $f$ through $p$, denoted ${\mathrm{Rift}}_{p}f$, is a 1-cell $\tilde{f}:A\to B$ equipped with a 2-cell

$$\epsilon :p\circ \tilde{f}\Rightarrow f$$\varepsilon: p \circ \widetilde{f} \Rightarrow f

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

$$\zeta :g\Rightarrow \tilde{f}$$\zeta: g \Rightarrow \widetilde{f}

such that $\epsilon •(p\circ \zeta )=\eta$. (Here $\circ$ refers to composition across a 0-cell, and $•$ to composition across a 1-cell.) As with any universal description, the pair $({\mathrm{Rift}}_{p}f,\epsilon )$ is unique up to unique 2-cell isomorphism.

A left Kan lift of $f$ through $p$, denoted ${\mathrm{Lift}}_{p}f$, is a 1-cell $\tilde{f}:A\to B$ equipped with a 2-cell

$$\eta :f\Rightarrow p\circ \tilde{f}$$\eta: f \Rightarrow p \circ \widetilde{f}

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

$$\zeta :\tilde{f}\Rightarrow g$$\zeta: \widetilde{f} \Rightarrow g

such that $(p\circ \zeta )•\eta =\theta$.

Global description

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

$${\mathrm{Rift}}_p:[A,C]\to [A,B]$$Rift_p: [A, C] \to [A, B]

which at the object level is of course $f\mapsto {\mathrm{Rift}}_{p}f$. At the morphism level, given a 2-cell

$$\eta :f\Rightarrow g:A\to C$$\eta: f \Rightarrow g: A \to C

there is an induced 2-cell ${\mathrm{Rift}}_{p}f\Rightarrow {\mathrm{Rift}}_{p}g$, the one which corresponds (by the universal property of ${\mathrm{Rift}}_{p}g$) to the composite

$$p\circ {\mathrm{Rift}}_{p}f\stackrel{\epsilon }{\Rightarrow }f\stackrel{\eta }{\Rightarrow }g.$$p \circ Rift_p f \overset{\varepsilon}{\Rightarrow} f \overset{\eta}{\Rightarrow} g.

A standard universality argument shows that ${\mathrm{Rift}}_p$ thus defined is functorial.

Proposition: ${\mathrm{Rift}}_p$ is right adjoint to the functor $[A,p]:[A,B]\to [A,C]$ obtained by postcomposing with $p$, i.e., the functor $f\mapsto p\circ f$.

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 $p$ itself has a left or right adjoint. Other examples will be given in the section which follows.

Examples

• Let $\mathrm{Mod}$ denote the framed bicategory of small categories $C$, bimodules $r:C\to D$ (i.e., functors $r:D^{\mathrm{op}}\times C\to \mathrm{Set}$), 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.,

  $$({\mathrm{Rift}}_{r}s)(b,c)={\int }_{d:D}\mathrm{hom}(r(d,c),s(d,b))$$(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 $\mathrm{Rel}$, which corresponds to enrichment in $2$, the right Kan lift is essentially a universally quantified predicate of the form

$${\forall }_{d:D}r(d,c)\Rightarrow s(d,b)$$\forall_{d: D} r(d, c) \Rightarrow s(d, b)

(“for all $d$ satisfying condition $r$, we impose condition $s$”).

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

• In $\mathrm{Cat}$, if $A$ is small and $B$ is locally small, and if $F:A\to B$ is a functor, then we have a Yoneda embedding $y:A\to PA={\mathrm{Set}}^{A^{\mathrm{op}}}$ and a functor $B(F-,-):B\to PA$, and there is a canonical map

  $$y_A\to B(F-,-)\circ f$$y_A \to B(F-, -) \circ f

  (essentially, $\mathrm{hom}(a,b)\to \mathrm{hom}(Fa,Fb)$ taking $f:a\to b$ taking $f:a\to b$ to $Ff:Fa\to Fb$). This arrow exhibits $F$ as a left Kan lift of $y$ through $B(F-,-)$. 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.