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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Cauchy complete category} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{idempotents}{}\paragraph*{{Idempotents}}\label{idempotents} [[!include idempotents - contents]] \hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory} [[!include category theory - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{InOrdinaryCategoryTheory}{In ordinary category theory}\dotfill \pageref*{InOrdinaryCategoryTheory} \linebreak \noindent\hyperlink{in_terms_of_splitting_of_idempotents}{In terms of splitting of idempotents}\dotfill \pageref*{in_terms_of_splitting_of_idempotents} \linebreak \noindent\hyperlink{InTermsOfTinyObjects}{In terms of tiny objects}\dotfill \pageref*{InTermsOfTinyObjects} \linebreak \noindent\hyperlink{InTermsOfAbsoluteColimits}{In terms of absolute colimits}\dotfill \pageref*{InTermsOfAbsoluteColimits} \linebreak \noindent\hyperlink{InOrdinaryCatTheoryByProfunctors}{In terms of profunctors}\dotfill \pageref*{InOrdinaryCatTheoryByProfunctors} \linebreak \noindent\hyperlink{InOrdinaryCatTheoryByEssGeomMorphisms}{In terms of essential geometric morphisms}\dotfill \pageref*{InOrdinaryCatTheoryByEssGeomMorphisms} \linebreak \noindent\hyperlink{InEnrichedCategoryTheory}{In enriched category theory}\dotfill \pageref*{InEnrichedCategoryTheory} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{metric_spaces}{Metric spaces}\dotfill \pageref*{metric_spaces} \linebreak \noindent\hyperlink{ordinary_enriched_categories}{Ordinary ($Set$-enriched) categories}\dotfill \pageref*{ordinary_enriched_categories} \linebreak \noindent\hyperlink{of_prosets}{Of prosets}\dotfill \pageref*{of_prosets} \linebreak \noindent\hyperlink{discussion}{Discussion}\dotfill \pageref*{discussion} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The concept of \emph{[[Cauchy-complete space|Cauchy completeness]]}, ordinarily thought of as applying to [[metric spaces]], was vastly generalized by [[Bill Lawvere]] in his influential paper \emph{\hyperlink{Lawvere}{Metric spaces, generalized logic, and closed categories}}. It is now seen by [[category theory|category theorists]] as a concept of [[enriched category theory]], with close ties to the concept of [[Morita equivalence]] in the theory of [[modules]]. In category theory one also speaks of \emph{[[idempotent completion|idempotent completeness]]}. The basic idea is that the Cauchy [[completion]] of a [[category]] is the closure of a category under what are called ``[[absolute limit]]s'', i.e., those [[limit]]s that are preserved by any [[functor]] whatsoever. Equivalently, the Cauchy completion is the closure with respect to [[absolute colimit]]s. If $C$ is [[small category|small]], the Cauchy completion $\bar{C}$ of $C$ lies between $C$ and its ``[[free cocompletion]]'', aka [[presheaf category]] \begin{displaymath} C \hookrightarrow \bar{C} \hookrightarrow Set^{C^{op}} \end{displaymath} and consists of the presheaves $F$ dubbed [[tiny object|tiny]] by Lawvere, meaning those presheaves which are [[connected object|connected]] and [[projective object|projective]]: the functor \begin{displaymath} hom_{Set^{C^{op}}}(F, -): Set^{C^{op}} \to Set \end{displaymath} preserves small [[coproducts]] and [[coequalizers]]. All of these concepts generalize straightforwardly to the context of general $V$-[[enriched category theory|enriched categories]], where $V$ is a [[complete category|complete]], [[cocomplete category|cocomplete]], [[symmetric monoidal category|symmetric]] [[monoidal closed category]]. Lawvere defines a \textbf{point} of the Cauchy completion of a small $V$-category $C$ to be a $V$-enriched [[bimodule]] $p: \mathbf{1} \to C$ (in other words, a $V$-functor $\mathbf{1} \to V^{C^{op}}$) for which there is a bimodule $q: C \to \mathbf{1}$ right adjoint to $p$ (in the [[bicategory]] of enriched bimodules, see [[profunctor]]), where $\mathbf{1}$ is the unit $V$-category. Thus points of the Cauchy completion are certain $V$-enriched presheaves $p: C^{op} \to V$, and together form a $V$-category called the \textbf{Cauchy completion} whose [[hom-object|hom]]s are the presheaf homs. It is denoted $\bar{C}$. As we will explain in more detail below, [[representable functor|representable presheaves]] belong to the Cauchy completion, and so the [[Yoneda embedding]] of $C$ factors through a full embedding \begin{displaymath} i: C \to \bar{C} \end{displaymath} and we say the $V$-category $C$ is \textbf{Cauchy-complete} if this embedding is an equivalence. We work through a few examples in the following section. \hypertarget{InOrdinaryCategoryTheory}{}\subsection*{{In ordinary category theory}}\label{InOrdinaryCategoryTheory} \hypertarget{in_terms_of_splitting_of_idempotents}{}\subsubsection*{{In terms of splitting of idempotents}}\label{in_terms_of_splitting_of_idempotents} \begin{defn} \label{CauchyCompletionByRetractsOfRepresentablePresheaves}\hypertarget{CauchyCompletionByRetractsOfRepresentablePresheaves}{} For $C$ a [[small category]] write \begin{displaymath} \overline{C} \hookrightarrow [C^{op}, Set] \end{displaymath} for the [[full subcategory]] of the [[category of presheaves]] on $C$ on the [[retract]]s of [[representable functor]]. This $\overline{C}$ is called the \textbf{Cauchy completion} of $C$. \end{defn} For instance (\hyperlink{BorceuxDejean}{BorceuxDejean, below theorem 1}). \begin{theorem} \label{}\hypertarget{}{} The Cauchy completion $\overline{C}$ satisfies the following properties \begin{enumerate}% \item $\overline{C}$ is a [[small category]]; \item $C$ is a [[full subcategory]] $C \hookrightarrow \overline{C}$; \item every [[idempotent]] in $\overline{C}$ [[split idempotent|splits]]; \item the inclusion $C \hookrightarrow \overline{C}$ is an [[equivalence of categories]] precisely if already every [[idempotent]] in $C$ splits; \item there is an [[equivalence of categories]] \begin{displaymath} [C^{op}, Set] \simeq [\overline{C}^{op}, Set] \,. \end{displaymath} \end{enumerate} \end{theorem} This appears for instance as (\hyperlink{BorceuxDejean}{BorceuxDejean, theorem 1}). \begin{proof} $\overline{C}$ is small because $[C^{op}, Set]$ is a [[well-powered category]]. It contains $C$ as a [[full subcategory]] because the [[Yoneda embedding]] is a [[full and faithful functor]]. Every idempotent splits in $\overline{C}$ because it does so in $[C^{op}, Set]$ and because the composite of two retractions is a retraction. A retract of a representable $y(c) \in [C^{op}, Set]$ induces an idempotent on $y(c)$ and hence by the [[Yoneda lemma]] an idempotent on $c \in C$. If $C$ is already idempotent complete, this splits and produces a retraction of $c$ in $C$ and hence of $y(C)$ in $[C^{op}, Set]$. Since this is necessarily isomorphic to the original retraction, we find that every retract of the representable $y(c)$ is itself representable, therefore $C \simeq \overline{C}$ in this case. \end{proof} \hypertarget{InTermsOfTinyObjects}{}\subsubsection*{{In terms of tiny objects}}\label{InTermsOfTinyObjects} \begin{prop} \label{CauchyComplIsFullSubcatOnTinyObjects}\hypertarget{CauchyComplIsFullSubcatOnTinyObjects}{} The Cauchy completion $\overline{C}$ is equivalently the [[full subcategory]] of $[C^{op}, Set]$ on the [[tiny object]]s (``small projective objects''). \end{prop} This appears for instance as (\hyperlink{BorceuxDejean}{BorceuxDejean, prop. 2}). \hypertarget{InTermsOfAbsoluteColimits}{}\subsubsection*{{In terms of absolute colimits}}\label{InTermsOfAbsoluteColimits} \begin{prop} \label{}\hypertarget{}{} The following conditions are equivalent for a [[small category]] $C$. \begin{enumerate}% \item $C$ is Cauchy complete; \item $C$ has all small [[absolute colimit]]s. \end{enumerate} \end{prop} \hypertarget{InOrdinaryCatTheoryByProfunctors}{}\subsubsection*{{In terms of profunctors}}\label{InOrdinaryCatTheoryByProfunctors} We discuss Cauchy completion of [[small categories]] in terms of [[profunctor]]s. Write $*$ for the terminal category (single object, single morphism). Let $C$ be a [[small category]] \begin{remark} \label{}\hypertarget{}{} The category $C$ is equivalent to the [[functor category]] out of the point \begin{displaymath} C \simeq Func(*,C) \,. \end{displaymath} Its [[category of presheaves]] is equivalent to the profunctor category \begin{displaymath} [C^{op}, Set] \simeq Profunc(*, C) \,. \end{displaymath} In these terms the [[Yoneda embedding]] $C \hookrightarrow [C^{op}, Set]$ is the canonical inclusion \begin{displaymath} Func(*,C) \hookrightarrow Profunc(*,C) \,. \end{displaymath} Accordingly the Cauchy completion, def. \ref{CauchyCompletionByRetractsOfRepresentablePresheaves} is a [[full subcategory]] of the profunctor category \begin{displaymath} \overline{C} \hookrightarrow Profunc(*,C) \,. \end{displaymath} \end{remark} \begin{prop} \label{CuachyCompByLeftAdjointProfunctor}\hypertarget{CuachyCompByLeftAdjointProfunctor}{} A [[profunctor]] $F : * ⇸ C$ belongs to $\overline{C}$ precisely if it has a [[right adjoint]] in [[Prof]]. \end{prop} This appears for instance as (\hyperlink{BorceuxDejean}{BorceuxDejean, prop. 4}). \begin{theorem} \label{}\hypertarget{}{} For a [[small category]] $C$, the following are equivalent \begin{enumerate}% \item $C$ is Cauchy complete; \item a profunctor $* ⇸ C$ has a [[right adjoint]] precisely if it is a functor; \item for every [[small category]] $A$ a profunctor $A ⇸ C$ has a [[right adjoint]] precisely if it is a functor; \end{enumerate} \end{theorem} This appears for instance as (\hyperlink{BorceuxDejean}{BorceuxDejean, theorem 2}). \hypertarget{InOrdinaryCatTheoryByEssGeomMorphisms}{}\subsubsection*{{In terms of essential geometric morphisms}}\label{InOrdinaryCatTheoryByEssGeomMorphisms} In the context of [[topos theory]] we say, for $C$ [[small category]], that an [[adjoint triple]] of [[functor]]s \begin{displaymath} Set \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}} [C,Set] \end{displaymath} is an [[essential geometric morphism]] of [[topos]]es $f : Set \to [C,Set]$; or an \textbf{[[point of a topos|essential point]]} of $[C,Set]$. By the [[adjoint functor theorem]] this is equivalently simply a single functor $f^* : [C, Set] \to Set$ that preserves all small [[limit]]s and [[colimit]]s. Write \begin{displaymath} Topos_{ess}(Set,[C,Set]) \simeq LRFunc([C,Set], Set) \subset Func([C,Set], Set) \end{displaymath} for the [[full subcategory]] of the [[functor category]] on functors that have a [[left adjoint]] and a [[right adjoint]]. \begin{prop} \label{CauchyCompletionByEssentialPoints}\hypertarget{CauchyCompletionByEssentialPoints}{} For $C$ a [[small category]] there is an [[equivalence of categories]] \begin{displaymath} \overline{C} \simeq Topos_{ess}(Set, [C,Set])^{op} \end{displaymath} of its Cauchy completion, def. \ref{CauchyCompletionByRetractsOfRepresentablePresheaves}, with the category of essential points of $[C,Set]$. \end{prop} \begin{proof} We first exhibit a [[full subcategory|full inclusion]] $Topos_{ess}(Set,[C,Set])^{op} \hookrightarrow \overline{C}$. So let $Set \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}} [C,Set]$ be an [[essential geometric morphism]]. Then because $f_!$ is [[left adjoint]] and thus preserves all small [[colimits]] and because every [[set]] $S \in Set$ is the colimit over itself of the singleton set we have that \begin{displaymath} f_! S \simeq \coprod_{s \in S} f_!(*) \end{displaymath} is fixed by a choice of [[copresheaf]] \begin{displaymath} F := f_!(*) \in [C, Set] \,. \end{displaymath} The $(f_! \dashv f^*)$-[[adjunction]] [[isomorphism]] then implies that for all $H \in [C,Set]$ we have \begin{displaymath} f^* H \simeq Set(*, f^* H) \simeq [C,Set](f_! *, H) \simeq [C,Set](F,H) \,. \end{displaymath} naturally in $H$, and hence that \begin{displaymath} f^*(-) \simeq [C,Set](F,-) : Set \to [C,Set] \,. \end{displaymath} By assumption this has a further right adjoint $f_\ast$ and hence preserves all [[colimits]]. By the discussion at [[tiny object]] it follows that $F \in [C,Set]$ is a tiny object. By prop. \ref{CauchyComplIsFullSubcatOnTinyObjects} this means that $F$ belongs to $\overline{C} \subset [C,Set]$. A morphism $f \Rightarrow g$ between [[geometric morphisms]] $f,g : Set \to [C,Set]$ is a [[geometric transformation]], which is a [[natural transformation]] $f^* \Rightarrow g^*$, hence by the above a natural transformation $[C,Set](F,-) \Rightarrow [C,Set](G,-)$. By the [[Yoneda lemma]] these are in bijection with morphisms $G \to H$ in $[C,Set]$. This gives the full inclusion $Topos_{ess}(Set,[C,Set])^{op} \subset \overline{C}$. The converse inclusion is now immediate by the same arguments: since the objects in $\overline{C}$ are precisely the [[tiny object]]s $F \in [C,Set]$ each of them corresponds to a functor $[C,Set](F,-) : [C,Set] \to Set$ that has a [[right adjoint]]. Since this generally also has a left adjoint, it is the [[inverse image]] of an essential geometric morphism $f : Set \to [C,Set]$. \end{proof} Write $Cat_{Cauchy} \hookrightarrow$ [[Cat]] for the [[full sub-2-category]] of [[Cat]] on the Cauchy-complete categories. \begin{prop} \label{}\hypertarget{}{} The 2-category of Cauchy complete categories is a [[coreflective subcategory|coreflective]] [[full sub-2-category]] of [[Topos]] with essential geometric morphisms \begin{displaymath} Cat_{Cauchy} \hookrightarrow Topos_{ess} \end{displaymath} exhibited by the [[2-adjunction]] \begin{displaymath} ([-,Set] \dashv Topos_{ess}(Set,-) ) : Topos_{ess} \stackrel{\overset{Cat_{Cauchy}(-,Set)}{\hookleftarrow}}{\underset{Topos_{ess}(Set,-)}{\to}} Cat_{Cauchy} \,. \end{displaymath} \end{prop} \begin{proof} We first claim that when working with all categories instead of just the Cauchy-complete categories there is a [[2-adjunction]] \begin{displaymath} ([-,Set] \dashv Topos_{ess}(Set,-) ) : Topos_{ess} \stackrel{\overset{[-,Set]}{\leftarrow}}{\underset{Topos_{ess}(Set,-)}{\to}} Cat \,. \end{displaymath} This is exhibited by the following [[equivalence of categories|equivalence]] of [[hom-categories]] \begin{displaymath} \begin{aligned} Func(C, Topos_{ess}(Set, E)) & \simeq Func(C, LRFunc(E, Set)) \\ & \simeq LRFunc(E, Func(C,Set)) =: LRFunc(E, [C,Set]) \\ & \simeq Topos_{ess}([C,Set], E) \end{aligned} \end{displaymath} [[natural equivalence|natural]] in $C \in Cat$ and $E \in Topos$. Here \begin{itemize}% \item the first equivalence is by definition of [[essential geometric morphism]]; \item the second equivalence follows by observing that limits and colimits in presheaf categories are computed objectwise; \item the third equivalence is again the definition of essential geometric morphisms. \end{itemize} Now by prop. \ref{CauchyCompletionByEssentialPoints} we have that the components of the [[unit of an adjunction|unit]] of this adjunction \begin{displaymath} C \to Topos_{ess}(Set,[C,Set]) \end{displaymath} are [[equivalence of categories|equivalences]] precisely if $C$ is Cauchy-complete. This means that restricted along $Cat_{Cauchy} \hookrightarrow Cat$ the adjunction exhibits a coreflective embedding. \end{proof} \hypertarget{InEnrichedCategoryTheory}{}\subsection*{{In enriched category theory}}\label{InEnrichedCategoryTheory} We discuss Cauchy completion in $\mathcal{V}$-[[enriched category]] theory, for $\mathcal{V}$ a [[closed category|closed]] [[symmetric monoidal category]] with all [[limits]] and [[colimits]]. The discussion in ordinary category theory \href{InOrdinaryCategoryTheory}{above} is the special case where $\mathcal{V} :=$ [[Set]]. The key to the enriched version is the reformulation of ordinary Cauchy completion in terms of [[profunctors]] as discussed \href{InOrdinaryCatTheoryByProfunctors}{above}. These have an immediate generalization to enriched category theory, and so one takes this formulation as the definition. As before, we have \begin{remark} \label{}\hypertarget{}{} Every small $\mathcal{V}$-category $C$ is equivalent to the $\mathcal{V}$-[[enriched functor category]] \begin{displaymath} C \simeq \mathcal{V}Func(I,C) \,, \end{displaymath} where $I$ is the $\mathcal{V}$-category with a single [[object]] $*$ and $I(*,*) = I_{\mathcal{V}}$, the tensor unit in $\mathcal{V}$. Also, \begin{displaymath} [C^{op}, \mathcal{V}] \simeq \mathcal{V}Profunc(I, C) \end{displaymath} and the canonical $\mathcal{V}$-[[enriched functor]] \begin{displaymath} \mathcal{V}Func(I,C) \hookrightarrow \mathcal{V}Profunc(I,C) \end{displaymath} is the enriched [[Yoneda embedding]]. \end{remark} \begin{defn} \label{}\hypertarget{}{} For $C$ a small $\mathcal{V}$-[[enriched category]], the \textbf{Cauchy completion} of $C$ is the full $\mathcal{V}$-subcategory \begin{displaymath} \overline{C} \hookrightarrow \mathcal{V}Profunc(I,C) \end{displaymath} on those [[profunctor]]s with a [[right adjoint]] in $\mathcal{V}$[[Prof]]. \end{defn} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \begin{itemize}% \item When $\mathcal{V} = \mathbf{Set}$, a $\mathcal{V}$-category is an ordinary category. The Cauchy completion of an ordinary category is its \emph{idempotent completion}, or [[Karoubi envelope]]. This also holds when $\mathcal{V} = \mathbf{Cat}$ or $\mathcal{V} = \mathbf{sSet}$, or more generally whenever $\mathcal{V}$ is a cartesian [[cosmos]] where the terminal object is [[tiny]]. \item When $\mathcal{V} = [0,\infty]$ is the extended nonnegative reals ordered by $\geq$ and with $+$ as monoidal product, $\mathcal{V}$-categories are generalized metric spaces. The Cauchy completion is the usual completion under \emph{Cauchy sequences}. \item When $\mathcal{V} = \mathbf{Ab}$ is abelian groups, a $\mathcal{V}$-category is a [[pre-additive category]]. The Cauchy completion is the completion under \emph{finite direct sums and idempotent splitting}. Notice that there is also a ``sub-Cauchy completion'' given by completing just under finite direct sums, which turns a pre-additive category into an [[additive category]]. \item When $\mathcal{V} = \mathbf{Slat}$ is the category of sup-lattices, a $\mathcal{V}$-category is a locally posetal, [[locally cocomplete bicategory]], i.e. a [[quantaloid]]. The Cauchy completion is some sort of completion under \emph{arbitrary} sums: it is large even if the original quantaloid is small, and its existence depends on the precise definition we choose of Cauchy completion. \item In the $\infty$-categorical context, we can consider enrichment in the $\infty$-category of [[spectra]]. The Cauchy completion of an $\infty$-category enriched in spectra is its completion under \emph{all finite colimits}. \item Generalizing to bicategorical enrichment, we can construct from a [[site]] $(\mathcal{C}, J)$ a certain bicategory $\mathcal{W}$ such that the Cauchy-complete, symmetric, skeletal $\mathcal{W}$-categories are just the sheaves on $(\mathcal{C}, J)$. Variations on this theme can yield $\mathcal{C}$-indexed categories, stacks, prestacks, or presheaves as Cauchy completions or sub-Cauchy completions for categories enriched in certain bicategories. \end{itemize} Now we look at two examples in more detail: metric spaces and ordinary categories. \hypertarget{metric_spaces}{}\subsubsection*{{Metric spaces}}\label{metric_spaces} We consider first the classical case of [[metric space]]s, but as redefined by Lawvere to mean a category enriched in the [[poset]] $V = ([0, \infty], \geq)$, with [[tensor product]] given by addition. So, to say $X$ is a [[Lawvere metric space]] means that with the set $X$ there is a distance function \begin{displaymath} d_X = hom_X: X \times X \to [0, \infty] \end{displaymath} such that \begin{displaymath} d(x, y) + d(y, z) \geq d(x, z), \qquad 0 \geq d(x, x) \end{displaymath} for all $x, y, z$ in $X$. (The associativity and identity axioms are here superfluous since $V$ is a poset.) A $V$-enriched functor $f: X \to Y$ here just means a function from $X$ to $Y$ such that \begin{displaymath} d_X(x, y) \geq d_Y(f(x), f(y)) \end{displaymath} for all $x, y$ in $X$ (again, preservation of composition and of identities is superfluous here), so that $V$-functors are [[short maps]] between metric spaces (Lipschitz maps with constant at most $1$). Finally, a $V$-enriched transformation $f \to g: X \to Y$ in this case boils down to an instance of a property: that \begin{displaymath} 0 \geq d_Y(f(x), g(x)) \end{displaymath} for all $x$ in $X$. If $f, g$ are valued in $[0, \infty]$, this just means $f(x) \geq g(x)$ for all $x$. A point of the Cauchy completion is an $X$-module $p: \mathbf{1} \to X$, i.e., an enriched functor or short map \begin{displaymath} p: X^{op} \to [0, \infty] \end{displaymath} for which there is an $X$-module $q: X \to \mathbf{1}$ on the other side, an enriched functor \begin{displaymath} q: X \to [0, \infty] \end{displaymath} that is [[right adjoint]] to $p$ in the sense of modules. This means there is a unit of the adjunction in the bicategory of modules: \begin{displaymath} (Id: \mathbf{1} \to \mathbf{1}) \to (\mathbf{1} \overset{p}{\to} X \overset{q}{\to} \mathbf{1}) \end{displaymath} and a counit: \begin{displaymath} (X \overset{q}{\to} \mathbf{1} \overset{p}{\to} X) \to (Id: X \to X) \end{displaymath} Recall now that $Id_X: X \to X$ in the bicategory of modules is the unit bimodule $y_X: X \to V^{X^{op}}$ given by the enriched [[Yoneda embedding]], or in different words, $hom_X = d_X: X^{op} \times X \to [0, \infty]$. Recall also that module composition is defined by a [[coend]] formula for a tensor product. If one now tracks through the definitions, keeping in mind that we are in the very simple case of enrichment in a poset, the unit of the [[adjunction]] $p \dashv q$ boils down to having the property \begin{displaymath} 0 \geq \int^{x \in X} q(x) + p(x) = \inf_{x \in X} q(x) + p(x) \end{displaymath} and the counit boils down to having the property \begin{displaymath} p(x) + q(y) \geq d_X(x, y) \end{displaymath} To better appreciate what these conditions mean, we point out that $p(x)$ should be thought of as the distance $d_{\bar{X}}(x, p)$ between $x$ and the ``ideal point'' $p$ in the Cauchy completion $\bar{X}$, and $q(x)$ should be thought of as the companion distance $d_{\bar{X}}(p, x)$. Thus the unit condition above would come down to saying that for every $\varepsilon \gt 0$ there exists $x \in X$ such that \begin{displaymath} d_{\bar{X}}(p, x) \lt \varepsilon, \, d_{\bar{X}}(x, p) \lt \varepsilon \end{displaymath} and the counit condition imposes a necessary triangle inequality constraint on the distance functions $p$ and $q$, in order that we get an actual Lawvere metric space $\bar{X}$. If $p, p'$ are two points of the Cauchy completion thus defined, then their distance is defined by the usual formula for enriched presheaves: \begin{displaymath} d(p, p') = \int_{x \in X} \hom_{[0, \infty]}(p(x), p'(x)) = \sup_{x \in X} \max\{0, p'(x) - p(x)\} \end{displaymath} It should be noted that even under the classical definition (where we impose symmetry $d(x, y) = d(y, x)$, separation $d(x, y) \gt 0$ for $x \neq y$, and finiteness $d(x,y) \lt \infty$), this provides an elegant alternative definition of Cauchy completion. In essence, all it is doing is taking the metric closure $\bar{X}$ of the embedding of $X$ into the already complete space of short maps: \begin{displaymath} y_X: X \to [0, \infty]^{X^{op}}: x \mapsto d_X(-, x) \end{displaymath} The presheaf-hom definition of the distance formula for $\bar{X}$, being manifestly non-symmetric, is not the usual definition of distance in the classical symmetric case. However, if we first symmetrize the distance in $[0, \infty]$: \begin{displaymath} \sigma d(r, s) = d(r, s) + d(s, r) = \max\{0, s-r\} + \max\{0, r-s\} = |r - s| \end{displaymath} or equivalently \begin{displaymath} \sigma d(r, s) = d(r, s) + d(s, r) = \max(\max\{0, s-r\}, \max\{0, r-s\}) = |r - s| \end{displaymath} then we do retrieve the classical formula \begin{displaymath} d(p, p') = \int_{x \in X} \sigma d(p(x), p'(x)) = \sup_{x \in X} |p(x) - p'(x)| \end{displaymath} In other words, the completion $\bar{X}$ of a symmetric metric space $X$ as a general (Lawvere) metric space is not necessarily the same as its completion $\sigma\bar{X}$ as a symmetric metric space, but $\sigma\bar{X}$ is the symmetrisation of $\bar{X}$. \hypertarget{ordinary_enriched_categories}{}\subsubsection*{{Ordinary ($Set$-enriched) categories}}\label{ordinary_enriched_categories} The analysis of Cauchy complete Lawvere metric spaces contains some of the seeds of what happens in other [[enriched category]] contexts; the case of ordinary [[small category|small categories]], where the enrichment is no longer in a mere [[poset]] but in [[Set]], reflects still more of the phenomena generally associated with Cauchy completions. Let $C$ be a [[small category]] and let the [[module]] $p: \mathbf{1} \to C$ be a point of $\bar{C}$, with module $q: C \to \mathbf{1}$ as its [[right adjoint]] in the [[bicategory]] of modules. As [[functor]]s, \begin{displaymath} p: C^{op} \to Set, \, q: C \to Set \end{displaymath} and the structure of the [[adjunction]] is given by unit and counit maps: \begin{displaymath} \eta: 1 \to \int^{c \in Ob(C)} q(c) \times p(c), \qquad \varepsilon_{c, d}: p(c) \times q(d) \to C(c, d) \end{displaymath} As we said in the case of metric spaces, $p(c)$ and $q(c)$ measure ``distances'' = homs: \begin{displaymath} p(c) \cong Set^{C^{op}}(C(-, c), p), \qquad q(c) \cong Set^{C^{op}}(p, C(-, c)) \end{displaymath} The first [[isomorphism]] is an instance of the [[Yoneda lemma]], and the second can be seen as follows. The set $q(c)$ is the [[bimodule]] composite \begin{displaymath} (\mathbf{1} \overset{c}{\to} C \overset{q}{\to} \mathbf{1}) \end{displaymath} where $c$ is shorthand for the module $C(-, c): C^{op} \to Set$; this is just an instance of the [[Yoneda lemma]]: \begin{displaymath} q \circ_C C(-, c) \overset{def}{=} \int^{d \in C} q(d) \times C(d, c) \overset{Yoneda}{\cong} q(c). \end{displaymath} Now using the adjunction $p \dashv q$, there are, for any set $S$, natural bijections \begin{displaymath} \frac{\frac{S \to q(c)}{S \to q \circ_C C(-, c)}}{\frac{p \circ_{\mathbf{1}} S \to C(-, c)}{p(-) \times S \to C(-, c)}} \end{displaymath} and maps in the bottom line are in bijection with maps $S \to Set^{C^{op}}(p, C(-, c))$. Therefore we have a natural bijection \begin{displaymath} \frac{S \to q(c)}{S \to Set^{C^{op}}(p, C(-, c))} \end{displaymath} and this proves $q(c) \cong Set^{C^{op}}(p, C(-, c))$. With these identifications of $q(c)$ and $p(c)$, the unit of the adjunction $p \dashv q$ takes the form \begin{displaymath} \eta: 1 \to \int^{c} Set^{C^{op}}(p, C(-, c)) \times Set^{C^{op}}(C(-, c), p) \end{displaymath} The [[coend]] above is a quotient of \begin{displaymath} \sum_c Set^{C^{op}}(p, C(-, c)) \times Set^{C^{op}}(C(-, c), p) \end{displaymath} and hence the unit element $\eta$ is represented by a pair of [[natural transformation|transformation]]s \begin{displaymath} i: p \to C(-, c), \qquad \pi: C(-, c) \to p \end{displaymath} for some $c$. Given that, it is now not hard -- in fact it is fairly tautological -- to verify that on the basis of the triangular equation of the adjunction which says \begin{displaymath} (p \overset{p \circ \eta}{\to} p \circ q \circ p \overset{\varepsilon}{\to} p) = 1_p, \end{displaymath} that \begin{displaymath} (p \overset{i}{\to} C(-, c) \overset{\pi}{\to} p) = 1_p \end{displaymath} and so a point $p$ in the Cauchy completion $\bar{C}$ must be a [[retract]] of a [[representable functor|representable]] $C(-, c)$. Spelling this out a little more: the composite \begin{displaymath} C(-, c) \overset{\pi}{\to} p \overset{i}{\to} C(-, c) \end{displaymath} is an [[idempotent]] represented by a morphism $e: c \to c$ in $C$ (by the Yoneda lemma), and this factorization through $p$ [[split idempotent|splits]] the idempotent $C(-, e)$ in $Set^{C^{op}}$. Indeed, the claim is that modules $p: C^{op} \to Set$ in the Cauchy completion are precisely those presheaves on $C$ which arise as [[retract]]s of [[representable functor|representable]]s in $Set^{C^{op}}$, or in other words may be identified with objects of the [[split idempotent|idempotent-splitting]] completion of $C$ (aka the \emph{[[Karoubi envelope]]} of $C$). Therefore, in the $Set$-enriched case, the Cauchy completion \emph{is} the idempotent-splitting completion. In particular, representables themselves are points of the Cauchy completion. Notice that in a finitely complete category (such as $Set$ or a [[presheaf]] category), idempotents $e: c \to c$ split automatically: just take the equalizer of the pair \begin{displaymath} c \stackrel{\overset{e}{\to}}{\underset{1}{\to}} c \end{displaymath} For that matter, in any finitely cocomplete category, taking the [[coequalizer]] of the above pair would also split the idempotent. Indeed, we can say that idempotents split in a category iff all equalizers of such pairs exist, iff all coequalizers of such pairs exist. Notice that if $C$ and $D$ are categories, then any functor $F: C \to D$ preserves [[retract]]s and therefore splittings of idempotents. Thus, the equalizers above are the sort of limits which are preserved by \emph{any} functor $F$ whatsoever. They are called \textbf{absolute limits} for that reason. For the same reason, the coequalizers above are \textbf{absolute colimits}: they are precisely the colimits preserved by any functor whatsoever. Pursuing this a bit further: if $F: C^{op} \to Set$ is any functor, then (because idempotents split in $Set$) there is a unique extension $\bar{F}: \bar{C}^{op} \to Set$ of $F$. Therefore we have an equivalence \begin{displaymath} Set^{C^{op}} \simeq Set^{\bar{C}^{op}} \end{displaymath} and we say that $C$ and $\bar{C}$ are \textbf{Morita equivalent}. \hypertarget{of_prosets}{}\subsubsection*{{Of prosets}}\label{of_prosets} Every ordinary [[poset]] clearly is Cauchy complete, since the only idempotents are the [[identity morphism]]s. The [[internalization]] of this statement requires some extra assumptions: \begin{prop} \label{}\hypertarget{}{} [[internalization|Internal]] to any [[regular category]] every [[poset]] is Cauchy complete. \end{prop} This appears as (\hyperlink{Rosolini}{Rosolini, prop. 2.1}). \begin{prop} \label{}\hypertarget{}{} [[internalization|Internal to]] any [[exact category]] the Cauchy completion of any [[preorder]] exists and is its [[poset reflection]]. \end{prop} This appears as (\hyperlink{Rosolini}{Rosolini, corollary. 2.3}). Moreover, the characterization of Cauchy completion by left adjoint profunctors requires the internal [[axiom of choice]]: \begin{prop} \label{}\hypertarget{}{} In a given ambient context, the following are equivalent: \begin{enumerate}% \item the [[axiom of choice]] holds; \item every [[profunctor]] $F : A ⇸ C$ between [[posets]] is an ordinary functor when it has a [[right adjoint]]. \end{enumerate} \end{prop} For instance (\hyperlink{BorceuxDejean}{BorceuxDejean, prop. 5}). \hypertarget{discussion}{}\subsection*{{Discussion}}\label{discussion} \emph{David}: Concerning the result that on Set the terminal F-coalgebra is the Cauchy completion of the initial F-algebra, for certain F, I wonder if we have to factor completions through the metric space completion, as Barr does in . Perhaps Adamek's work on is more natural. Does this all tie in with the [[ideal completion]] as by Awodey where you sum types/sets in a topos into a universal object? How many kinds of completion are there for an enriched category? I see some may coincide in certain cases. If two categories can be Morita equivalent, should this be reflected in the page [[Morita equivalence]]? \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[split idempotent]] \item [[Karoubi envelope]] \item [[idempotent complete (infinity,1)-category]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} The notion of Cauchy complete categories was introduced in \begin{itemize}% \item [[Bill Lawvere]], \emph{Metric spaces, generalized logic and closed categories} Rend. Sem. Mat. Fis. Milano, 43:135--166 (1973) Reprints in Theory and Applications of Categories, No. 1 (2002) pp 1-37 (\href{http://www.tac.mta.ca/tac/reprints/articles/1/tr1abs.html}{tac}) \end{itemize} Surveys are in \begin{itemize}% \item [[Francis Borceux]] and D. Dejean, \emph{Cauchy completion in category theory} Cahiers Topologie G\'e{}om. Diff\'e{}rentielle Cat\'e{}goriques, 27:133--146, (1986) (\href{http://www.numdam.org/item?id=CTGDC_1986__27_2_133_0}{numdam}) \item A. Carboni and [[Ross Street]], \emph{Order ideal in categories} Pacific J. Math., 124:275--288, 1986. \end{itemize} Further references include for instance \begin{itemize}% \item S. R. Johnson, \emph{Small Cauchy Completions} , JPAA \textbf{62} (1989) pp.35-45. \item R. Walters, \emph{Sheaves and Cauchy complete categories} , Cahiers Top. Geom. Diff. Cat. 22 no. 3 (1981) 283-286 (\href{http://www.numdam.org/item?id=CTGDC_1981__22_3_283_0}{numdam}) \item R. Walters, \emph{Sheaves on sites as Cauchy-complete categories, J. Pure Appl. Algebra 24 (1982) 95-102} \end{itemize} Cauchy completion of [[internalization|internal]] [[prosets]] is discussed in \begin{itemize}% \item G. Rosolini, \emph{A note on Cauchy completeness for preorders} (\href{http://www.disi.unige.it/person/RosoliniG/notccp.pdf}{pdf}) \end{itemize} [[!redirects Cauchy complete category]] [[!redirects Cauchy complete categories]] [[!redirects Cauchy-complete category]] [[!redirects Cauchy-complete categories]] [[!redirects Cauchy complete]] [[!redirects Cauchy-complete]] [[!redirects Cauchy completeness]] [[!redirects Cauchy-completeness]] [[!redirects Cauchy completion]] [[!redirects Cauchy completions]] [[!redirects Cauchy-completion]] [[!redirects Cauchy-completions]] [[!redirects idempotent complete category]] [[!redirects idempotent complete categories]] [[!redirects idempotent-complete category]] [[!redirects idempotent-complete categories]] \end{document}