nLab Kan lift

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

Of course, lifts typically don’t literally exist in the sense of an equation $p \circ \widetilde{f} = f$ or an isomorphism $p \circ \widetilde{f} \cong f$. But in good situations, one may have the next best thing: a 2-cell $p \circ \widetilde{f} \Rightarrow f$ 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 $p: B \to C$, $f: A \to C$ in a bicategory, a right Kan lift of $f$ through $p$, denoted $Rift_p f$, is a 1-cell $\widetilde{f}: A \to B$ equipped with a 2-cell

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

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

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

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

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

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

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

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

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

Global description

Given $p: B \to C$ and a 0-cell $A$ in a bicategory, if the right Kan lift $Rift_p f$ 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

$$Rift_p: [A, C] \to [A, B]$$

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

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

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

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

A standard universality argument shows that $Rift_p$ thus defined is functorial.

Proposition: $Rift_p$ 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 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 $(\widetilde{f}, \varepsilon) = \mathop{Rift}_p f$, $g$ is said to respect this right lift if $(\widetilde{f}g, \varepsilon \bullet g) = \mathop{Rift}_p(f g)$

and analogously for left Kan lifts. A Kan lift $\widehat{f}$ is absolute if it is respected by any 1-cell into $\mathop{dom}(\widehat{f})$. 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 $Mod$ denote the framed bicategory of small categories $C$, bimodules $r: C \to D$ (i.e., functors $r: D^{op} \times C \to Set$), 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.,
  $$(Rift_r s)(b, c) = \int_{d: D} hom(r(d, c), s(d, b))$$

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

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

(“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.

• adjunctions in a 2-category can be defined in terms of Kan lifts: a 1-cell $u\colon A \to B$ has a left adjoint iff $\mathop{Lift}_u 1_B$ exists and is absolute; in this case putting $(f,\iota) = \mathop{Lift}_u 1_B$ we have $f \dashv u$ with unit $\iota \colon 1_B \Rightarrow u f$. 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 in $\mathbf{Cat}$ can also be expressed as absolute kan lifts; see relative adjoint for a precise statement.

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

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

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

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

Related concepts

• relative adjunction

References

Kan lifts were mentioned in passing in Definition I.12.6 of:

• John Gray, Formal category theory: adjointness for $2$-categories, Lecture Notes in Mathematics, 391, Springer (1974) [doi:10.1007/BFb0061280]

where it is mentioned the concept does not have a name since they had never been considered before in Cat.

Kan lifts were called “liftings” in:

• Ross Street, Bob Walters, Yoneda structures on 2-categories, JPAA 50 (1978) 350-379 [doi:10.1016/0021-8693(78)90160-6]

The terminology “Kan lifting” is used in:

• Charles Grellois, Algebraic theories, monads, and arities (2011) [arXiv:1110.3294, hal:00670841]

The terminology “Kan lift” seems to first appear in:

• Weng Kin Ho, Characterising E-projectives via Co-monads, Electronic Notes in Theoretical Computer Science 301 (2014) 61-77 [doi:10.1016/j.entcs.2014.01.006]

In Remark 5.9 of the following paper, it is observed that the pointwise analogue of Kan lifts (corresponding to the pointwise notion of Kan extension) is precisely the notion of relative adjunction:

• Nathanael Arkor, Dylan McDermott, The formal theory of relative monads, Journal of Pure and Applied Algebra 107676. (2024) [arXiv:2302.14014&rbrack.