# Category where every Morphism is a Mono

Is there a well-known term for a category where, equivalently:

1. every morphism is a monomorphism
2. every slice category is a preorder

?

• I've bumped into the exact same structure in my research. I did not find suitable/universally accepted terminology. I had to invent it myself. – Randall Dec 5 '17 at 18:16
• And, what was your term? – Berci Dec 5 '17 at 22:49
• 1. is not equivalent to 2., right? Sure the skeleton of each slice is a preorder, but you can have many isomorphic and nonequal fields. – Fosco Dec 6 '17 at 8:59
• A preorder seen as a category is usually called thin. A category such that each slice has proprerty $P$ is often called a locally $P$ category. So maybe a locally thin category? – Pece Dec 6 '17 at 13:11
• @FoscoLoregian , preorder is the same as thin except I think for size issues. You might be thinking of poset which is a preorder with anti-symmetry. – Max New Dec 6 '17 at 16:51

## 1 Answer

A category in which there is at most one morphism between two objects is called thin. A category for which each slice is a gizmo category is often called a locally guizmo category (for example locally cartesian closed category).

Hence it makes sense to call a category in which every morphism is mono a locally thin category.

Beware though that it is not standard and I wouldn't use it without redefining it first.