Given an object$X$ in some category, a family $(f_i\colon U_i \to X)_i$ of morphisms to $X$ is an epic sink, or a jointly epic family if, given any two morphisms $g, h\colon X \to Y$ such that $g \circ f_i= h\circ f_i$ for all $i$, it follows that $g = h$.

Dually, a family $(f_i\colon X \to U_i)_i$ of morphisms from $X$ is a monic source, or a jointly monic family if, given any two morphisms $g, h\colon Y \to X$ such that $f_i \circ g = f_i \circ h$ for all $i$, it follows that $g = h$.

Sometimes we are interested only in small families of morphisms, but if so then it is best to say so explicitly.

A single morphism $U \to X$ is an epimorphism if and only it forms an epic sink by itself; conversely, a sink $(f_i\colon U_i \to X)_i$ is epic iff the induced map $\coprod_i U_i \to X$ is an epimorphism, assuming that the coproduct$\coprod_i U_i$ exists. (Note, though, that for a large family of morphisms, this coproduct might not exist even if the category has all small coproducts.) Dual results hold for monomorphisms and products.

If a functor $F \colon J \to C$ has a colimit $\mathrm{colim}F$, with maps $\iota_i \colon F i \to \mathrm{colim}F$ for $i \in J$, then the family $(\iota_i)_{i \in J}$ is jointly epic. Similarly, the maps $(\pi_i \colon \mathrm{lim}F \to F i)_{i\in J}$ are jointly monic, when the limit exists.

Last revised on April 21, 2021 at 13:50:47.
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