A sink is a pair ((fi)i∈I, A), sometimes denoted by (fi,A)I or (Ai →fi A)I consisting of an object A (the codomain of the sink) and a family of morphisms fi : Ai → A indexed by some class I. The family (Ai)i∈I is called the domain of the sink. Composition of sinks is defined in the (obvious) way dual to that of composition of sources.
In Set, a sink (Ai →fi A)I is an epi-sink if and only if it is jointly surjective, i.e., iff A = ∪i∈I fi[Ai]. In every construct, all jointly surjective sinks are epi-sinks. The converse implication holds, e.g., in Vec, Pos, Top, and Σ-Seq. A category A is thin if and only if every sink in A is an epi-sink.
Every epi-sink (= jointly surjective sink) is an extremal epi-sink in Set, Vec, and Ab. In Top an epi-sink (Ai →fi A) is extremal if and only if A carries the final topology fi with respect to (fi)i∈I. In Pos an epi-sink (Ai →fi A)I is extremal iff the ordering of A is the transitive closure of the relation consisting of all pairs (fi(x), fi(y)) with i ∈ I and x ≤ y in Ai. In Σ-Seq an epi-sink (Ai →fi A)I is extremal iff each final state of A has the form fi(q) for some i ∈ I and some final state q of Ai.
Every separator is extremal in Set, Vec, and Ab. In Pos the separators are precisely the nonempty posets, whereas the extremal separators are precisely the non-discrete posets. Top has no extremal separator.