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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*{Rel} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{relations}{}\paragraph*{{Relations}}\label{relations} [[!include relations - contents]] \hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory} [[!include category theory - contents]] \hypertarget{categories_of_categories}{}\paragraph*{{Categories of categories}}\label{categories_of_categories} [[!include categories of categories - contents]] \hypertarget{}{}\section*{{$Rel$}}\label{} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{relations_and_spans}{Relations and spans}\dotfill \pageref*{relations_and_spans} \linebreak \noindent\hyperlink{limits_and_colimits}{Limits and colimits}\dotfill \pageref*{limits_and_colimits} \linebreak \noindent\hyperlink{weak_equalizers_and_completion}{Weak equalizers and completion}\dotfill \pageref*{weak_equalizers_and_completion} \linebreak \noindent\hyperlink{in_the_double_category_of_relations}{In the double category of relations}\dotfill \pageref*{in_the_double_category_of_relations} \linebreak \noindent\hyperlink{as_an_enriched_category}{As an enriched category}\dotfill \pageref*{as_an_enriched_category} \linebreak \noindent\hyperlink{in_spans}{In spans}\dotfill \pageref*{in_spans} \linebreak \noindent\hyperlink{monoepimorphisms}{Mono/epimorphisms}\dotfill \pageref*{monoepimorphisms} \linebreak \noindent\hyperlink{2categorical_aspects}{2-categorical aspects}\dotfill \pageref*{2categorical_aspects} \linebreak \noindent\hyperlink{relations_in_a_category}{Relations in a category}\dotfill \pageref*{relations_in_a_category} \linebreak \noindent\hyperlink{related_categories}{Related categories}\dotfill \pageref*{related_categories} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} $Rel$ is the [[category]] whose [[objects]] are [[sets]] and whose [[morphisms]] are (binary) [[relations]] between sets. It becomes a [[2-category]] $\mathbf{Rel}$ (in fact, a [[2-poset]]) by taking [[2-morphisms]] to be inclusions of relations. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} $Rel$ is a category whose [[objects]] or $0$-cells are [[sets]], whose [[morphisms]] or $1$-cells $X \to Y$ are [[relations]] $R \subseteq X \times Y$, and where the \emph{composite} $S \circ R$ of morphisms $R: X \to Y$ and $S: Y \to Z$ is defined as the usual relational composite: \begin{displaymath} \{(x, z) \in X \times Z: \exists_{y \in Y} (x, y) \in R \wedge (y, z) \in S\} \subseteq X \times Z .$$ The _identity_ $1_X: X \to X$ is then the [[equality relation]], in other words the usual diagonal embedding $$\{(x, x): x \in X\} \subseteq X \times X. \end{displaymath} Another important operation on relations is taking the \emph{opposite}, also known as the \emph{transpose} or sometimes the \emph{converse}. Any relation $R: X \to Y$ induces a relation \begin{displaymath} R^\dagger = \{(y, x) \in Y \times X: (x, y) \in R\} \subseteq Y \times X \end{displaymath} This makes $Rel$ into a [[dagger-category]]: in other words, we have $(S \circ R)^\dagger = R^\dagger \circ S^\dagger$, $1_X^\dagger = 1_X$, and $(R^{\dagger})^{\dagger} = R$. We can enhance $Rel$ to a 2-category, which we can call $\mathbf{Rel}$ to distinguish it from the mere category, by taking [[2-morphisms]] or $2$-cells $R \Rightarrow S$ to be inclusions of relations. Since there is at most one 2-morphism between parallel 1-morphisms, $\mathbf{Rel}$ is a [[2-poset]]: that is, an [[enriched category|category enriched]] in the category of [[posets]]. \hypertarget{relations_and_spans}{}\subsection*{{Relations and spans}}\label{relations_and_spans} It is useful to be aware of the connections between the [[bicategory of relations]] and the [[bicategory]] of [[span|spans]]. Recall that a \emph{span} from $X$ to $Y$ is a diagram of the form \begin{displaymath} X \leftarrow S \to Y \end{displaymath} and there is an obvious category whose objects are spans from $X$ to $Y$ and whose morphisms are morphisms between such diagrams. The terminal span from $X$ to $Y$ is \begin{displaymath} X \stackrel{\pi_1}{\leftarrow} X \times Y \stackrel{\pi_2}{\to} Y \end{displaymath} and a relation from $X$ to $Y$ is just a [[subobject]] of the terminal span, in other words an isomorphism class of monos into the terminal span. To each span $S$ from $X$ to $Y$, there is a corresponding relation from $X$ to $Y$, defined by taking the [[image]] of the unique morphism of spans $S \to X \times Y$ between $X$ and $Y$. It may be checked that this yields a lax morphism of bicategories \begin{displaymath} Span \to Rel \end{displaymath} \hypertarget{limits_and_colimits}{}\subsection*{{Limits and colimits}}\label{limits_and_colimits} The category $Rel$ does have [[products]] and [[coproducts]]; they coincide (by self-duality) and are just [[disjoint unions]] of sets. However, otherwise $Rel$ has very few (co)limits; it doesn't even have splittings of all idempotents. All symmetric idempotents have splittings, but the order-relation $\leq \; \subseteq \{0,1\} \times \{0,1\}$ can't be split. It follows that it can't have (co)equalisers. Since the category $Rel$ is the category of \emph{free algebras} ([[Kleisli category]]) for the powerset monad, there is, indeed, very little chance of a limit of such algebras being free again. To get decent limits, one has to move to the [[Eilenberg-Moore category]] of the [[powerset]] [[monad]], viz., the category of complete [[suplattices]]. \hypertarget{weak_equalizers_and_completion}{}\subsubsection*{{Weak equalizers and completion}}\label{weak_equalizers_and_completion} As the category $Rel$ has \emph{[[weak limit|weak]]} [[equalizers]], one can take its [[exact completion]]. This happens to be the category of complete sup-lattices and sup-preserving maps. And the tensor product on $Rel$ extends to the [[exact completion]]. The Freyd completion of $Rel$, which is equivalent to the category of basic pairs which appear in formal topology, has all limits exactly because $Rel$ has products and weak equalizers. The Freyd completion adds freely a strong factorization system to a(ny) category $C$. The Freyd completion has products if $C$ has products, and it has equalizers if $C$ has weak equalizers. \hypertarget{in_the_double_category_of_relations}{}\subsubsection*{{In the double category of relations}}\label{in_the_double_category_of_relations} If you insert the category $Rel$ into the [[double category]] $\mathrm{RRel}$ of sets, mappings and relations, one has a double category with all [[double limits]] and colimits. For instance, the obvious cartesian product $a \times b : X \times Y \to X' \times Y'$ (resp. sum $a+b : X + Y \to X' + Y'$) of two relations is indeed a product (resp. a sum) in the double category. Similarly, many bicategories of spans, cospans, relations, profunctors, etc. have poor (co)limits, but can be usefully embedded in weak double categories (with the same objects, ``strict morphisms'', ``same morphisms'', suitable double cells) that have all limits and colimits. \hypertarget{as_an_enriched_category}{}\subsection*{{As an enriched category}}\label{as_an_enriched_category} Writing $\mathcal{V}$ for the category of [[suplattices]], $Rel$ is a $\mathcal{V}$-category (see [[enriched category]]). With that in mind, the parallel: \begin{itemize}% \item in [[additive categories]] with [[zero object]], \emph{finitary} (co)[[products]] are [[biproducts]] \item $\mathcal{V}$-categories with [[zero object]], \emph{arbitrary} (co)[[products]] are [[biproducts]] \end{itemize} is remarkable. This was probably first noticed by [[Dana May Latch]], in the 20th century, for the category of $\mathcal{V}$ of complete [[suplattices]]. The [[matrix algebra]] for maps to products, from coproducts, and most especially, from coproducts to products, works just as it does in the case of [[additive categories]], when it comes to these $\mathcal{V}$-categories. \hypertarget{in_spans}{}\subsection*{{In spans}}\label{in_spans} See [[van Kampen colimit]]. \hypertarget{monoepimorphisms}{}\subsection*{{Mono/epimorphisms}}\label{monoepimorphisms} It is not hard to see that a relation $R \subseteq A \times B$ is a [[monomorphism]] $A \to B$ iff the map $\mathcal{P}A \to \mathcal{P}B$ sending a subset of A to the set of all R-relatives of its members is injective; dually for [[epimorphisms]]. \hypertarget{2categorical_aspects}{}\subsection*{{2-categorical aspects}}\label{2categorical_aspects} Recall that we are using $\mathbf{Rel}$ to stand for the 2-category of sets, relations and inclusions. Various facts about relations can be recast in these terms. For example, a relation $L : X \to Y$ is a function from $X$ to $Y$ iff it has a right adjoint $R : Y \to X$ in $\mathbf{Rel}$. In this case, its right adjoint equals its transpose $L^\dagger : Y \to X$. Similarly, a monad $T : X \to X$ in the 2-category $\mathbf{Rel}$ is exactly a preorder on $X$, since the existence of a unit $i: 1_X \Rightarrow T$ says that $T$ is reflexive and the existence of the multiplication $m: T^2 \Rightarrow T$ says that $T$ is transitive. This monad has an [[Eilenberg-Moore category|Eilenberg-Moore object]] iff it is an equivalence relation, in which case the Eilenberg-Moore object is the set of equivalence classes of the relation. \hypertarget{relations_in_a_category}{}\subsection*{{Relations in a category}}\label{relations_in_a_category} More generally, given any [[regular category]] $C$, one can form a 2-category of relations $\mathbf{Rel}(C)$ in similar fashion. The objects of $\mathbf{Rel}(C)$ are objects of $C$, the morphisms $r: c \to d$ in $\mathbf{Rel}(C)$ are defined to be [[subobject|subobjects]] of the terminal span from $c$ to $d$, and 2-cells $r \to s$ are subobject inclusions. To form the composite of $r \subseteq c \times d$ and $s \subseteq d \times e$, one takes the [[image]] of the unique span morphism \begin{displaymath} r \times_c s \to c \times e \end{displaymath} in the category of spans from $c$ to $e$, thus giving a mono into the terminal span from $c$ to $e$. The subobject class of this mono defines the relation \begin{displaymath} s \circ r \subseteq c \times e \end{displaymath} and the axioms of a regular category ensure that $\mathbf{Rel}(C)$ is a 2-category with desirable properties. Similar to what was said above, there is again a lax morphism of bicategories \begin{displaymath} Span(C) \to \mathbf{Rel}(C) \end{displaymath} There is also a functor \begin{displaymath} i: C \to \mathbf{Rel}(C) \end{displaymath} that takes a morphism $f: c \to d$ to the functional relation defined by $f$, i.e., the relation defined by the subobject class of the mono \begin{displaymath} \langle 1, f\rangle: c \to c \times d \end{displaymath} Such functional relations may also be characterized as precisely those 1-cells in $Rel(C)$ which are [[adjunction|left adjoints]]; the right adjoint of $\langle 1, f \rangle$ is the opposite relation $\langle f, 1\rangle$. The unit amounts to a condition \begin{displaymath} 1_c \subseteq \langle f, 1 \rangle \circ \langle 1, f \rangle \end{displaymath} which says that the functional relation is \emph{total}, and the counit amounts to a condition \begin{displaymath} \langle 1, f \rangle \circ \langle f, 1 \rangle \subseteq 1_d \end{displaymath} which says the functional relation is \emph{well-defined}. \hypertarget{related_categories}{}\subsection*{{Related categories}}\label{related_categories} \begin{itemize}% \item [[Set]] \end{itemize} A [[category of correspondences]] is a generalization of a category of relations. The composition of relations is that of correspondences followed by [[(-1)-truncated|(-1)-truncation]]. For generalizations of $Rel$ see \begin{itemize}% \item [[allegory]], [[cartesian bicategory]] \item [[bicategory of relations]] \item [[Prof]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item Grandis, M., Pare, R. :\href{https://eudml.org/doc/91618}{Limits in double categories}. Cahiers Topologie Géom. Différentielle Catég. 40(3), 162--220 (1999) \item Bucalo, A., Rosolini, G., Completions, comonoids, and topological spaces Annals of Pure and Applied Logic 137, 104-125 (2006) \item \emph{Credits for the section on limits go to the contributors of the following threads on the categories list:} \href{http://thread.gmane.org/gmane.science.mathematics.categories/8027}{categories: Limits and colimits in Rel?}, \href{http://thread.gmane.org/gmane.science.mathematics.categories/8186}{categories: Limits in REL} \end{itemize} category: category [[!redirects Rel]] [[!redirects REL]] [[!redirects rel]] \end{document}