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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> colimit 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]] =-- #### Limits and colimits +--{: .hide} [[!include infinity-limits - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea The concept of _colimit_ is that [[duality|dual]] to a [[limit]]: a colimit of a [[diagram]] in a [[category]] is, if it exists, the [[representable functor|co-classifying space]] for morphisms _out_ of that diagram. The intuitive general idea of a colimit is that it defines an object obtained by sewing together the objects of the diagram, according to the instructions given by the morphisms of the diagram. We have * the notion of _colimit_ generalizes the notion of [[direct sum]]; * the notion of _[[weighted colimit]]_ generalizes the notion of _weighted (direct) sum_. Sometimes colimits (or some colimits) are called _[[inductive limits]]_ or _[[direct limits]]_; see the discussion of terminology at [[limit]]. A [[weighted colimit]] in $C$ is a [[weighted limit]] in $C^{op}$. ## Definition A [[colimit]] in a [[category]] $C$ is the same as a [[limit]] in the [[opposite category]], $C^{op}$. More in detail, for $F : D^{op} \to C^{op}$ a functor, its [[limit]] $\lim F$ is the colimit of $F^{op} : D \to C$. ## Examples Here are some important examples of colimits: * A colimit of the [[diagram|empty diagram]] is an [[initial object]]. * A colimit of a diagram consisting of two (or more) objects and no nontrivial morphisms is their [[coproduct]]. * A colimit of a [[span]] is a [[pushout]]. * A colimit of two (or more) [[parallel morphisms]] is a [[coequalizer]]. * A colimit of a diagram whose domain is a [[sifted category]] is a [[sifted colimit]]. * A colimit of a diagram whose domain is a [[filtered category]] is a [[filtered colimit]]. * A colimit of a connected diagram is a [[connected colimit]]. * A colimit of a nonempty diagram is a [[nonempty colimit]]. See also *[[limits and colimits by example]]*. ## Properties The properties of colimits are of course [[formal duality|dual]] to those of [[limits]]. It is still worthwhile to make some of them explicit: \begin{proposition} \label{ColimitsInTermsOfCoequalizers} All colimits may be expressed via [[coequalizers]] of maps between [[coproducts]]. \end{proposition} This was historically first observed by [Maranda 1962, Thm. 1](#Maranda62). See the [[formal dual|dual]] discussion ([here](limit#ConstructionFromProductsAndEqualizers)) of [[limits]] via [[products]] and [[equalizers]]. \begin{proposition} **([[hom-functor preserves limits|contravariant Hom sends colimits to limits]])** \linebreak For $C$ a [[locally small]] category, for $F : D \to C$ a functor, for $c \in C$ an object and writing $C(F(-), c) : D \to Set$, we have $$ C(colim F, c) \simeq lim C(F(-), c) \,. $$ \end{proposition} Depending on how one introduces limits this holds by definition or is an easy consequence. In fact, this is just rewriting the fact that the covariant Hom respects [[limit]]s (as described there) in $C^{op}$ in terms of $C$: $$ \begin{aligned} C(colim F, c) & \simeq C^{op}(c, colim F) \\ & \simeq C^{op}(c, lim F^{op}) \\ & \simeq lim C^{op}(c, F^{op}(-)) \\ & \simeq lim C(F(-), c) \end{aligned} $$ Notice that this actually says that $C(-,-) : C^{op} \times C \to Set$ is a [[continuous functor]] in both variables: in the first it sends limits in $C^{op}$ and hence equivalently colimits in $C$ to limits in $Set$. \begin{proposition} **([[adjoints preserve (co-)limits|left adjoint functors preserve colimits]])** \linebreak Let $L : C \to C'$ be a functor that is [[left adjoint]] to some functor $R : C' \to C$. Let $D$ be a [[small category]] such that $C$ admits limits of shape $D$. Then $L$ commutes with $D$-shaped colimits in $C$ in that for $F : D \to C$ some diagram, we have $$ L(colim F) \simeq colim (L \circ F) \,. $$ \end{proposition} +-- {: .proof} ######Proof Using the adjunction isomorphism and the above fact that commutes with limits in both arguments, one obtains for every $c' \in C'$ $$ \begin{aligned} C'(L (colim F), c) & \simeq C(colim F, R(c')) \\ & \simeq lim C(F(-), R(c')) \\ & \simeq lim C'(L \circ F(-), c') \\ & \simeq C'(colim (L \circ F), c') \,. \end{aligned} \,. $$ Since this holds naturally for every $c'$, the [[Yoneda lemma|Yoneda lemma, corollary II]] on uniqueness of representing objects implies that $R (lim F) \simeq lim (R \circ F)$. =-- ## Related concepts * [[filtered colimit]] * [[directed colimit]] * [[sequential colimit]] * [[sifted colimit]] * [[direct sum]] * [[homotopy colimit]], [[(∞,1)-colimit]] * [[lax colimit]] ## References [[limit|Limits]] and colimits were defined in [[Daniel M. Kan]] in Chapter II of the paper that also defined [[adjoint functors]] and [[Kan extensions]]: * [[Daniel M. Kan]], _Adjoint functors_, Transactions of the American Mathematical Society 87:2 (1958), 294–294 ([doi:10.1090/s0002-9947-1958-0131451-0](https://doi.org/10.1090/s0002-9947-1958-0131451-0)). The observation that colimits may be constructed from [[coequalizers]] and set-indexed [[coproducts]]: * {#Maranda62} [[Jean-Marie Maranda]], Thm. 1 in: *Some remarks on limits in categories*, Canadian Mathematical Bulletin **5** 2 (1962) 133-146 &lbrack;[doi:10.4153/CMB-1962-015-0](https://doi.org/10.4153/CMB-1962-015-0)&rbrack; Beware that these early articles refer to colimits as _[[direct limits]]_. Textbook account: * [[Saunders MacLane]], §III.3 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (second ed. 1997) &lbrack;[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)&rbrack; [[!redirects colimits]] [[!redirects colimit functor]] [[!redirects colimit functors]] </textarea> </div>  </body> </html>