Category where every Morphism is a Mono Is there a well-known term for a category where, equivalently:


*

*every morphism is a monomorphism

*every slice category is a preorder


?
 A: 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.
A: The following are equivalent for a small category $\mathcal{C}$:

*

*Every morphism in $\mathcal{C}$ is a monomorphism.


*Every slice of $\mathcal{C}$ is a preorder category.


*There exists a surjective discrete fibration $\tilde{\mathcal{C}} \to \mathcal{C}$ where $\tilde{\mathcal{C}}$ is a preorder category.


*The presheaf topos $[\mathcal{C}^\textrm{op}, \textbf{Set}]$ is an étendue.
So you might call $\mathcal{C}$ an étendue category.
This is similar to how a coherent topos is one that admits a site of definition that is a coherent category.
(And note that every topos is a coherent category, so it matters whether "coherent" is being used as a modifier of "category" or "topos"!)
Or you might keep things simple and note that a one-object category has this property if and only if it corresponds to a left-cancellative monoid, so you might call it a left-cancellative category.
