subterminal object in nLab

<!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> subterminal object 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]] =-- =-- =-- #Contents# * table of contents {:toc} ## Definition +-- {: .num_defn} ###### Definition An [[object]] $U$ in a [[category]] $C$ is **subterminal** or **preterminal** if any two [[morphism]]s with [[target]] $U$ and the same source are equal. In other words, $U$ is subterminal if for any object $X$, there is at most one morphism $X\to U$. =-- +-- {: .num_defn} ###### Definition An __[[umbrella category]]__ is a nonempty category $C$ such that for every object $X$ in $C$, there is at least one subterminal object $T$ such that $C(X,T)$ is nonempty (hence being a singleton). =-- ## Properties If $C$ has a [[terminal object]] $1$, then $U$ is subterminal precisely if the unique morphism $U \to 1$ is [[monomorphism|monic]], so that $U$ represents a [[subobject]] of $1$; hence the name "sub-terminal." This is equivalent to the hypothesis that the [[cone]] given by identity morphisms $U \leftarrow U \rightarrow U$ is a [[product]] cone, or that *some* [[product]] $U \times U$ exists and the [[diagonal]] $U \to U \times U$ is an [[isomorphism]]. Therefore for a [[sheaf topos]] over a [[topological space]] the subterminal objects of the topos are the [[open subsets]] of the topological space. Accordingly, the subterminal objects in any [[topos]] are also called _open objects_ (e.g. [Johnstone 77, p. 94](#Johnstone77)) The [[classifying topos]] for subterminal objects (hence open objects) in [[toposes]] is the [[Sierpinski topos]] (see e.g. [Johnstone 77, p. 117](#Johnstone77)). ## Examples The subterminal objects in a [[topos]] can be viewed as its "external [[truth value]]s." For example, in the topos $Sh(X)$ of [[sheaf|sheaves]] on a [[topological space]] $X$, the subterminal objects are precisely the open sets in $X$. The *support* of an object $X$ in a topos is the [[image]] $U \hookrightarrow 1$ of the unique map $X \to 1$. Any map $U \to X$ is necessarily a [[section]] of $X \to U$. ## Related concepts * [[subsingleton]] * [[support object]] [[!include types and logic - table]] ## References * {#Johnstone77} [[Peter Johnstone]], _Topos theory_, London Math. Soc. Monographs __10__, Acad. Press 1977 * [[Dieter Pumpl眉n]], _Initial morphisms and monomorphisms_, Manuscripta mathematica 32 (1980): 309-333. [[!redirects subterminal]] [[!redirects subterminals]] [[!redirects subterminal objects]] [[!redirects preterminal]] [[!redirects preterminal object]] [[!redirects preterminal objects]] [[!redirects open object]] [[!redirects open objects]] </textarea> </div>  </body> </html>

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subterminal object in nLab