Example. Consider the category where all morphisms are . Not that these objects are numbers, but just that we can compare two elements. This is surprisingly a proper category. ![[Thin Category-1766816738146.png|218x213]] observe:
- no multiple arrows between two objects
- you don’t need to have an arrow between all objects, since this is not a total order. def. Thin Category (=Posetal category) is where all homsets have at most one morphism. thm. All partially ordered sets are a category. Thin category is equivalent to the above preorder category, as is the following subset category.