# 📝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

\begin{tikzcd} a \arrow[loop, "\leq", swap] \arrow[r, "\leq"] \arrow[rr, "\leq", bend right, swap] & b \arrow[r, "\leq"] & c \end{tikzcd}

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