Let $\mathcal{C}$ be a category, and $X$ an object in $\mathcal{C}$ such that for any other object $Y$, there exists a unique epimorphism $$ f: X \to Y $$ and this property identifies $X$ up to unique isomorphism. Do such objects have a name?
2026-03-25 12:46:41.1774442801
Name for a certain universal object in a category
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Such an object $X$ is simply an initial object in the wide subcategory of epimorphisms. A category with an initial object $0$ where all morphisms (or even just morphisms with domain $0$) are epic is necessarily a preorder.