# 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 pairs $\langle X,\rho \rangle$ (where $\rho$ is a relation on set $X$) and morphisms are relation-preserving functions. The standard category of sets and relations -apparently- is not mentioned anywhere.

Question: is there a standard name for the category described in ACC ?

• Yes, both categories are often denoted by $\Bbb{Rel}$, but in most cases they don't turn up in the same context. May 9, 2014 at 10:53
• The category you're considering is extensively studied in the book Categories, Allegories by Freyd and Ščedrov. May 9, 2014 at 11:09

Both categories are denoted by $\mathrm{Rel}$. In most cases, however, the author either explicitly tells which $\mathrm{Rel}$ he/she is using, or reproduces the definition of $\mathrm{Rel}$. (Note: this applies to any definition/term in math which can be used for two different things.) As egreg has mentioned in the comments, Categories, Allegories, Freyd and Ščedrov covers this.

Perhaps what you're looking for is a relation between $\mathrm{Rel}$ as defined in $\mathit{ACC}$, p. $22$ and $\mathrm{Rel}$ in the standard sense. If so, then note that $\mathrm{Rel}$ as defined in $\mathit{ACC}$, p. $22$ is $\mathrm{Rel}$ in the standard sense with an extra condition that the $2$-morphisms in $\mathrm{Rel}$ (in the standard sense) are relation-preserving maps, which are also morphisms in $\mathrm{Set}$, since the $2$-morphisms in $\mathrm{Rel}$ in the standard sense not only take relations $\rho\to\sigma$, but also takes the sets $X\to Y$.

• thank you Sanath. Indeed I was ultimately interested in the relation between the 2 categories. Could you perhaps express this relation just using the language of basic category theory (i.e. avoiding higher categories theory)? May 12, 2014 at 0:52
• @magma Yes - of course. If we label each relation as the vertex of a graph, then the mappings $X\to Y$ are the (directed) edges of the aforementioned graph.
– user122283
May 12, 2014 at 0:54