# Is there a special name for a category with all morphisms being morphisms from an initial object?

Let $\mathcal C$ be a (small) category whose only non-identity morphisms are morphisms from an initial object.

I guess $\mathcal C$ can be obtained by adjoining an initial object to a discrete category.

Is there a special name for this type of category? (I might have called it a "bouquet of morphisms", but this doesn't capture the orientation of the morphisms.)

• For 0, 1, 2, 3 objects these categories have names: 0, 1, $\to$, and span category. Is there a reason you'd expect them to have names in general? – Mees de Vries Jun 1 '18 at 9:30
• @MeesdeVries "span" is useful, thank you. I guess "expect" would be too strong, but I thought it is not unreasonable to ask if someone knows of a special name for these — you list names for 0–3, there is one answer mentioning "flat domain" and there is also "multispan" (which turns out to be more general than what I am looking for). – Earthliŋ Jun 1 '18 at 9:49