Category where every Morphism is a Mono

Question

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

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

?

Answer 1

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.

Answer 2

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.

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