3
votes
1answer
48 views

Rel: the category of relations

$\text{Rel}$ is the standard name for the category of sets and relations. Confusingly in "Abstract and concrete categories" (ACC), page 22, $\text{Rel}$ is defined as a category whose objects are ...
1
vote
1answer
52 views

Does $f \leq f \circ f^\dagger \circ f$ hold in an arbitrary allegory?

If $f$ is an arrow of $\mathrm{Rel}$, then $f \leq f \circ f^\dagger \circ f.$ Proof. Suppose $xy \in f$. Then $xy \in f, yx \in f^\dagger, xy \in f$. Thus $xy \in f \circ f^\dagger \circ f.$ ...
4
votes
0answers
65 views

Where can I learn more about the concept that is dual to “relation”?

Let $X$ and $Y$ denote sets. Then a relation $X \rightarrow Y$ is, by definition, a subset of $X \times Y$. Dually, we can define that a "corelation" $X \rightarrow Y$ is a partitioning of $X \uplus ...
8
votes
3answers
163 views

Can we take images of equivalence relations?

Given a function $f : X \rightarrow Y$, it is well-known that we can take the image under $f$ of any subset $A \subseteq X$, and we can take the preimage under $f$ of any subset $A \subseteq Y$. This ...
0
votes
0answers
41 views

Embedding vs restriction

Embedding is the morphism $( A ; B ; \operatorname{id}_A)$ of the category $\mathbf{Rel}$ for sets $A \subseteq B$. I call restriction the morphism $( A ; B ; \operatorname{id}_B)$ for sets $A ...
2
votes
1answer
47 views

Can objects repeat in commutative diagrams?

Are objects allowed to repeat in commutative diagrams? This seems to be necessary when representing endomorphisms such as the morphism $f : X \to X$ in the category $\mathbf{Set}$, such as when $f$ is ...
2
votes
0answers
84 views

How to denote the set of binary relations of which a particular ordered pair is a member?

Given a universe $U$ and two subsets $S$ and $T$ (also, both members of $U$), what is the name given to denote the set of all binary relations in $U$ where the ordered pair $(S,T)$ is a member? The ...
4
votes
1answer
57 views

How does one charaterize functionhood (etc.) in the category of relations?

Question: How can I complete the following sentence: "For all $\mathsf{Rel}$-arrows $f : X \rightarrow Y$, $f$ is a function iff ..."? Of course, this is easily done: "For all $\mathsf{Rel}$-arrows ...
3
votes
1answer
107 views

Mono's and Epi's in the category Rel?

Sorry to ask such a trivial question, but I can't find the answer anywhere. Question. What are the monomorphisms/epimorphisms in Rel? Furthermore, what's the standard terminology for describing ...
4
votes
0answers
84 views

Directed and projective limit in Rel

I'm looking for the directed (or equivalently projective) limit of a directed family of relations in the category of sets with relations between them. Consider a family $\{ R_{ij} \subseteq U_i ...
2
votes
2answers
413 views

Distinguishing equality and isomorphism as relations

Is this relational characterization of equality in Wikipedia accepted? The identity relation is the archetype of the more general concept of an equivalence relation on a set: those binary ...
1
vote
1answer
89 views

Categories of $n$-ary relations?

Arrows in the category $\bf Rel$ are binary (2-valued) relations between set objects. Do ternary, 4-term, $n$-term and variadic (2-valued) relations form categories? (Or perhaps one category?). It ...
0
votes
1answer
209 views

Direct products in the category Rel

Please describe direct products in the category Rel.
1
vote
0answers
42 views

Product of relations described in categorical terms?

Can cartesian product of several (not necessarily binary) relations be described in categorical terms? Don't propose me direct product in the category Set, as it does not preserve the structure of ...
6
votes
2answers
742 views

What is the difference between Categories and Relations?

For a common basis, I'll state basic definitions of a category and the relation type I'm thinking of. They're here for quick clarity, not precision, so feel free to revise for an answer. Category: A ...