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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> final lift in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="noindex,nofollow" /> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } </style> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script type="text/javascript"> <![CDATA[//><!]]> </script> </head> <body> <div id="Container"> <textarea id='content' readonly=' readonly' rows='24' cols='60' > +-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Category theory +-- {: .hide} [[!include category theory - contents]] =-- =-- =-- # Final and initial lifts * table of contents {: toc} ## Definition Let $U\colon C\to D$ be a [[functor]]. +-- {: .un_defn} ###### Definition A **$U$-structured sink** is a [[sink]] of the form $\{f_i \colon U(A_i) \to Y\}$ in $D$. =-- Note that like all sinks, a $U$-structured sink is not necessarily assumed to be [[small category|small]]. A **lift** of $Y$ along a $U$-structured sink $\{f_i\colon U(A_i)\to Y\}$ is an object $B$ of $C$, equipped with a sink $\{\phi_i\colon A_i \to B\}$ in $C$ and a morphism $h\colon Y\to U(B)$ in $D$, such that $U(\phi_i) = h \circ f_i$ for each $i$. A **morphism of lifts** is, of course, a morphism $\xi\colon B\to B'$ of $C$ such that the sink $\{\phi'_i\colon A_i\to B'\}$ factors through the sink $\{\phi_i\colon A_i\to B\}$ as $\phi_i'=\xi\circ\phi_i$, and such that the morphism $h'\colon Y\to U(B')$ factors through the morphism $h\colon Y\to U(B)$ as $h'=p(\xi)\circ h$. Note that if $U$ is faithful, then it suffices to demand merely that the sink $\{\phi_i\colon A_i \to B\}$ exists, rather than giving it as part of the structure. +-- {: .un_defn} ###### Definition A **semi-final lift** of $\{f_i \colon U(X_i) \to Y\}$ is a lift $B$ of $Y$ admitting a unique morphism of lifts to any other lift of $Y$. If the morphism $h\colon Y\to U(B)$ is an isomorphism, then the semi-final lift is called a **final lift**. If $h$ is an identity, we call $B$ a **strictly final lift**. =-- If objects of $C$ are regarded as objects of $D$ equipped with [[stuff, structure, property|structure]], for a (strictly) final lift we say that $B$ is the **final structure** or **strong structure** on $Y$ induced by the sink. Note that if $U$ is an [[isofibration]], then any final lift may be made into a strictly final one. The dual concept, which applies to cosinks ("sources"), is called a (perhaps semi- or strictly) **initial lift**, an **initial structure** or a **weak structure**. ## Examples * If $D$ is the [[terminal category]], then a $U$-structured sink is simply a family of objects $A_i$ of $C$, a lift is just a cocone $A_i\to B$, a morphism of lifts is a morphism of cocones, hence a semi-final lift is simply a [[colimit]] in $C$. * An empty $U$-structured sink is just an object $Y$ of $D$, a lift of it is a morphism $Y\to U(B)$, and a morphism of lifts is a morphism $B\to B'$ so that $Y\to U(B')$ factors as $Y\to U(B)\to U(B')$. Hence, a semi-final lift of such a sink is a [[free object]] with unit morphism $h\colon Y\to U(B)$ of $D$. Thus $U$ admits semi-final lifts of empty sinks precisely when it has a [[left adjoint]]. Similarly, it admits final lifts of empty sinks precisely when it has a [[fully faithful functor|fully faithful]] left adjoint (i.e. if it admits [[discrete objects]]). * A singleton $U$-structured sink is just a morphism of the form $f\colon U(X) \to Y$. A strictly final lift of such a sink is precisely an [[opcartesian arrow]] lying over $f$. Thus $U$ admits strictly final lifts of singleton structured sinks precisely when it is a [[Grothendieck opfibration]] (and final lifts of such sinks precisely when it is a [[Street opfibration]]). * A [[topological concrete category]] is a functor that admits final lifts of *all* (not necessarily small) structured sinks. This turns out to be equivalent to admitting initial lifts of all structured cosinks. The most famous example is then [[initial topology|initial topologies]] and [[final topology|final topologies]] for $U\colon Top \to Set$. * More generally, a [[solid functor]] is one that admits *semi-final* lifts of all structured sinks. * If $U$ has both a left and right adjoint, of which one (and hence also the other) is fully faithful, and $C$ is cocomplete, then $U$ admits final lifts of all *small* structured sinks. See [[adjoint triple]] for a proof. Dually, if $C$ is complete in this situation, then $U$ admits initial lifts of all small structured cosinks. ## Remarks * (Semi-)final lifts can be generalized to *(semi-)final extensions*, which are to (semi-)final lifts as [[Kan extensions]] are to [[colimits]]. * In [[Higher Topos Theory]] (section 4.3.1) the corresponding notion of (strictly) *final lift* for [[(∞,1)-categories]] is called a *$U$-colimit*. [[!redirects final lift]] [[!redirects final lifts]] [[!redirects semi-final lift]] [[!redirects semi-final lifts]] [[!redirects semifinal lift]] [[!redirects semifinal lifts]] [[!redirects strictly final lift]] [[!redirects strictly final lifts]] [[!redirects final structure]] [[!redirects final structures]] [[!redirects strong structure]] [[!redirects strong structures]] [[!redirects initial lift]] [[!redirects initial lifts]] [[!redirects semi-initial lift]] [[!redirects semi-initial lifts]] [[!redirects semiinitial lift]] [[!redirects semiinitial lifts]] [[!redirects strictly initial lift]] [[!redirects strictly initial lifts]] [[!redirects initial structure]] [[!redirects initial structures]] [[!redirects weak structure]] [[!redirects weak structures]] </textarea> </div>  </body> </html>