A notation for a morphism in a thin category

Question

Consider a thin category with objects $A\leq B$. There exists a unique morphism $A\rightarrow B$.

Is there a standard notation for this morphism (given $A$ and $B$)?

Answer 1

"$A \leq B$" is a pretty good name for it.

By analogy with posets of open sets in topological spaces, $i_{A,B}$ is good too.

Answer 2

Note that a thin category is nothing else than a $\{0,1\}$-enriched category. If we interpret $0$ as "false" and $1$ as "true", then a morphism $A \to B$ can be identified with the truth value of $A \leq B$. Hence, $A \leq B$ is a good notation for this unique morphism $A \to B$, and it coincides with the usual meaning.

Answer 3

There is no standard terminology for that, nor is there any terminology at all for that morphism. The category $Pos$ of posets is equivalent to the category of (small) categories with at most one morphisms between any two objects, but the categories are not isomorphic nor is there a canonical equivalence between the two categories. This is basically the situation you are describing. The reason there isn't any particular terminology for that morphism is that it is a lot more convenient to refer simply to $A\le B$ and leave the name of the morphism implicit. Which equivalence one chooses has no practical implications, and introducing any notation for the implicit morphism will only serve to clutter the notation.