Category of Pointed Relations

Question

There are the categories $\mathbf{Set}$ and $\mathbf{pSet}$; given the category $\mathbf{Rel}$, can we define an analogous category $\mathbf{pRel}$ as well? A relation $R \subseteq A \times B$ is said to be a morphism from $(A, a)$ to $(B, b)$ if $(a, b) \in R$. This should be compatible with relation composition and the identity relation. However, I can't find references on this - does this mean that somehow this category is uninteresting?

Answer 1

The category of pointed relations is considered in the literature in the context of logic and computer science (even the concept of multi pointed relations is considered). A google search for "pointed relations" will bring more than a handful of hits (though it is true that pointed relations are not widely used.

Quite often passing from a category to its pointed version is done in order to obtain required or desired categorical products (like having a zero object, or having a well-behaved tensor product). This is the case for $\bf {Rel}$. It is used often in the context of dagger categories where the pointed version is not needed. For some applications though the pointed version is used.