category theory / Mathematics Category-Theoretic Sinks May 16, 2017 AltExploitLeave a comment 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} →^{fi} 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} →^{fi} 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} →^{fi} 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} →^{fi} 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} →^{fi} 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}) →^{fi} (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}) →^{fi} (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}) →^{fi} (X, α))_{I} in Spa(T) is final iff α = ∪_{I} Tf_{i}[α_{i}]. AdvertisementProliferateTweetShare on TumblrWhatsAppMoreEmailLike this:Like Loading... Related