# Tagged Questions

1answer
40 views

### Terminology on pullbacks

I'm quite confused with the use of pullbacks, and in particular I wonder which terminology I shall use in the following examples. Let $X$ and $Y$ be arbitrary sets. Suppose that $f,g:X\to Y$ and I ...
1answer
51 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 ...
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 $xy \in f \circ f^\dagger \circ f.$ ...
0answers
65 views

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 ...
0answers
85 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 ...
1answer
59 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 ...
1answer
113 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 ...
0answers
84 views

1answer
212 views

### Direct products in the category Rel

Please describe direct products in the category Rel.
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 ...
2answers
752 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 ...