# 📝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.

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 $|C(a,b)| + |C(b,a)| \in \{0,1,2\}$