Category-Theoretic Sinks

The concept dual to that of source is called sink. Whereas the concepts of sources and sinks are dual to each other, frequently sources occur more naturally than sinks.

A sink is a pair ((f_i)_i∈I, A), sometimes denoted by (f_i,A)_I or (A_i →^f_i A)_I consisting of an object A (the codomain of the sink) and a family of morphisms f_i : A_i → A indexed by some class I. The family (A_i)_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 (A_i →^f_i A)_I is an epi-sink if and only if it is jointly surjective, i.e., iff A = ∪_i∈I f_i[A_i]. 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 (A_i →^f_i A) is extremal if and only if A carries the final topology f_i with respect to (f_i)_i∈I. In Pos an epi-sink (A_i →^f_i A)_I is extremal iff the ordering of A is the transitive closure of the relation consisting of all pairs (f_i(x), f_i(y)) with i ∈ I and x ≤ y in A_i. In Σ-Seq an epi-sink (A_i →^f_i A)_I is extremal iff each final state of A has the form f_i(q) for some i ∈ I and some final state q of A_i.

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.

An empty sink (∅, A) in a concrete category is final iff A is discrete.

A singleton sink A →^f B in a concrete category is final iff f is a final morphism.

A sink ((X_i, τ_i) →^f_i (X, τ))_I in Top is final iff τ is the final topology with respect to the maps (f_i)I, i.e. , τ = {U ⊆ X|∀ i ∈ I, f⁻¹[U] ∈ τ_i}.

A sink ((X_i, ≤_i) →^f_i (X, ≤))_I in Pos is final iff ρ is the transitive closure of the relation {(x, x)|x ∈ X} ∪ ∪_I {(f_i(x), f_i(y))|x ≤_i y}.

A sink ((X_i, α_i) →^f_i (X, α))_I in Spa(T) is final iff α = ∪_I Tf_i[α_i].

Advertisement