Talk:Category of relations

Binary relation?

Doesn't the word "relation" in this article precisely mean binary relation? If I am right, we need to specify this. Relation in general can have an arbitrary arity. --Acepectif 02:45, 15 May 2007 (UTC)

Hmm, no, the relations can be arbitrary spiders. User:Linas (talk) 19:50, 27 November 2013 (UTC)

Oh, I see the issue; its written as if it were a binary relation. The best of my understanding, an arbitrary relation can be built up out of tensoring with the 'cloning' and 'deleting' relations, and contracting the various 'spider legs' as needed. Basically, I think its just saying that an arbitrary relation and be built up out of binary relations: statements like "if this is that, and that is greater than the other, and the other is greater than B, then Q is less than S, etc." Some formalities, stating this, in the language of the units and counits of this cat, would be nice. (Hmm .. its vaguely similar to saying that a category that has all pushouts and all equalizers has all diagrams ... but in a simpler, less demanding setting. Curious... must explore more...) User:Linas (talk) 20:10, 27 November 2013 (UTC)

Relations as objects section - isn't this a separate category?

Most of this article describes a category where objects are sets and morphisms are relations, but the section "Relations as Objects" describes a category where the objects are (homogeneous) relations and the morphisms are relation-preserving maps.

It seems that these are actually two different categories, which happen to share the name "Rel". If this is the case it should be clearly stated that there are two different categories and the article describes both. (Unless they are isomorphic? But I don't think they are.) Nathaniel Virgo (talk) 15:30, 11 March 2020 (UTC)

Yes, the second definition has morphisms similar to an order isomorphism or a Galois connection, but based on a homogeneous relation. It is listed here, so far, as an alternative Rel, but with references it might be moved. The first definition, generalizing category of sets, includes heterogeneous relations as morphisms. — Rgdboer (talk) 20:19, 20 March 2020 (UTC)

Equivalence to the category of matrices over the booleans

Consider just finite sets and relations. The resulting category is equivalent to the category of finite-dimensional matrices over the boolean semiring. A lot of observations in this article are more intuitive in this case. Assuming the axiom of choice, the category Rel is equivalent to the category of matrices where the number of rows and columns are arbitrary cardinal numbers. Svennik (talk) 17:58, 29 March 2023 (UTC)