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\begin{document}

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\section*{Kan lift}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{2category_theory}{}\paragraph*{{2-Category theory}}\label{2category_theory}

[[!include 2-category theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{terminology}{Terminology}\dotfill \pageref*{terminology} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{local_description}{Local description}\dotfill \pageref*{local_description} \linebreak
\noindent\hyperlink{global_description}{Global description}\dotfill \pageref*{global_description} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The notion of \emph{Kan lift} in a [[2-category]] or in a [[bicategory]] is [[duality|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|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]]

\begin{displaymath}
\itexarray{
 & & B \\
 & & \downarrow p \\
A & \overset{f}{\to} & C
}
\end{displaymath}
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-morphism|2-cell]] $p \circ \widetilde{f} \Rightarrow f$ which is universal among 2-cells of this form. This gives the notion of \emph{right} Kan lift. The notion of \emph{left} Kan lift is similar, but with 2-cells in the opposite direction.

\hypertarget{terminology}{}\subsubsection*{{Terminology}}\label{terminology}

Usually, when working within a 2-categorical context, Kan lifts are simply refered to as \emph{lifts/liftings}, just as what happens with [[Kan extension|Kan extensions]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{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 functor]]s.

\hypertarget{local_description}{}\subsubsection*{{Local description}}\label{local_description}

Given 1-cells $p: B \to C$, $f: A \to C$ in a [[bicategory]], a \textbf{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

\begin{displaymath}
\varepsilon: p \circ \widetilde{f} \Rightarrow f
\end{displaymath}
satisfying the [[universal property]]: given any pair $(g: A \to B, \eta: p \circ g \Rightarrow f)$, there exists a unique 2-cell

\begin{displaymath}
\zeta: g \Rightarrow \widetilde{f}
\end{displaymath}
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 \textbf{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

\begin{displaymath}
\eta: f \Rightarrow p \circ \widetilde{f}
\end{displaymath}
such that given any pair $(g: A \to B, \theta: f \Rightarrow p \circ g)$, there exists a unique 2-cell

\begin{displaymath}
\zeta: \widetilde{f} \Rightarrow g
\end{displaymath}
such that $(p \circ \zeta) \bullet \eta = \theta$.

\hypertarget{global_description}{}\subsubsection*{{Global description}}\label{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 \textbf{global} Kan lift. When this is the case, we may define a functor between hom-categories

\begin{displaymath}
Rift_p: [A, C] \to [A, B]
\end{displaymath}
which at the object level is of course $f \mapsto Rift_p f$. At the morphism level, given a 2-cell

\begin{displaymath}
\eta: f \Rightarrow g: A \to C
\end{displaymath}
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

\begin{displaymath}
p \circ Rift_p f \overset{\varepsilon}{\Rightarrow} f \overset{\eta}{\Rightarrow} g.
\end{displaymath}
A standard universality argument shows that $Rift_p$ thus defined is functorial.

\textbf{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.

\begin{itemize}%
\item In a \href{http://golem.ph.utexas.edu/category/2009/06/kan_lifts.html}{post} at the nCaf\'e{}, [[David Corfield]] put the question: why are Kan extensions more frequently mentioned than Kan lifts? One proposed \href{http://golem.ph.utexas.edu/category/2009/06/kan_lifts.html#c024705}{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.

\end{itemize}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

Dual to the situation in Kan extensions, one is interested in whether a Kan lift is \emph{respected} by a 1-cell with codomain the domain of the lift. This is defined as follows:

\begin{itemize}%
\item given $(\widetilde{f}, \varepsilon) = \operatorname{Rift}_p f$, $g$ is said to \emph{respect} this right lift if $(\widetilde{f}g, \varepsilon \bullet g) = \operatorname{Rift}_p(f g)$

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

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item 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 limit]]s or via [[end]] formulas, e.g.,\begin{displaymath}
(Rift_r s)(b, c) = \int_{d: D} hom(r(d, c), s(d, b))
\end{displaymath}


\end{itemize}
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

\begin{displaymath}
\forall_{d: D} r(d, c) \Rightarrow s(d, b)
\end{displaymath}
(``for all $d$ satisfying condition $r$, we impose condition $s$'').

\begin{itemize}%
\item 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.


\item 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 $\operatorname{Lift}_u 1_B$ exists and is absolute; in this case putting $(f,\iota) = \operatorname{Lift}_u 1_B$ 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.


\item relative adjoints in $\mathbf{Cat}$ can also be expressed as absolute kan lifts; see [[relative adjoint]] for a precise statement.


\item 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 $(1_A, 1_f) = \operatorname{Lift}_f f$, and this lifting is absolute.


\item 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

\begin{displaymath}
y_A \to B(F-, -) \circ f
\end{displaymath}
(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 \href{http://www.pps.jussieu.fr/~weber/Two-toposes4.pdf}{development} in the context of 2-topos theory.



\end{itemize}
[[!redirects Kan lifting]]



\end{document}