nLab jointly epimorphic family

Definition

Given an object $X$ in some category $\mathcal{C}$, 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.

Finally, the empty family of morphisms with domain $X$ is a monic source iff $X$ is a subterminal object (and dually).

Warning: In the presence of coproducts in $\mathcal{C}$ the condition of being an jointly epic family can be expressed by the requirement that the induced map $\bigsqcup_{i} U_i \rightarrow X$ is epic. In the absence of coproducts it thus might be tempting to require the map $\bigsqcup_i h_{U_i} \rightarrow h_X$ to be an epic in $\operatorname{Psh}(\mathcal{C})$ or $\operatorname{Psh}_{\sqcup}(\mathcal{C})$, where the latter denotes the free coproduct completion. However this is incorrect and gives a stronger (i.e. sufficient) condition. The correct reformulation is that the induced map $\mathcal{C}(X,-)\rightarrow \prod_i \mathcal{C}(U_i,-)$ is a monomorphism of copresheaves.

Examples

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.

Related concepts

• cancellative category