📝Thin Category
Thin category or posetal category is a category where for every hom-set, it is either empty or a singleton set (one element). i.e., there is at most one arrow between any two objects per direction.
\begin{equation*} \begin{tikzcd} x \arrow[r, "f", shift left] \arrow[r, "g", shift right, swap] & y \end{tikzcd} \implies f = g \end{equation*}Thin category is isomorphic to preorder (see Order).
Preorder is a binary relation (≤) that is reflexive and transitive.
reflexivity gives the identity
transitivity gives the composition
Q:
Why is it called posetal category if it’s isomorphic to preorder (not poset)?
the requirement for hom-set to be empty or contain one element does not give antisymmetry as