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].

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Category Theoretic (Mono-)/Sources

A source is a pair (A,(f_i)_i∈I>) consisting of an object A and a family of morphisms f_i : A → A_i with domain A, indexed by some class I. A is called the domain of the source and the family (A_i)_i∈I is called the codomain of the source.

(1) Whenever convenient we use more concise notations, such as (A, f_i)_I, (A, f_i) or f_i

(A →^f_i A_i)_I.

(2)  The indexing class I of a source (A,f_i)_I may be a proper class, a nonempty set, or the empty set. In case I = ∅, the source is determined by A. In case I ≠ ∅, the source is determined by the family (f_i)_I.

(3)  Sources indexed by the empty set are called empty sources and are denoted by (A,∅). Whenever convenient, objects may be regarded as empty sources.

(4)  Sources that are indexed by a set are called set-indexed or small.

(5)  Sources that are indexed by the set {1, . . . , n} are called n-sources and are denoted by (A, (f₁, . . . , f_n)). Whenever convenient, morphisms f : A → B may be regarded as 1-sources (A,f).

(6)  There are properties of sources that depend heavily on the fact that (f_i)_I is a family, i.e., an indexed collection (e.g., the property of being a product). There are other properties of sources (A,f_i), depending on the domain A and the associated class {f_i|i ∈ I} only (e.g., the property of being a mono-source). In order to avoid a clumsy distinction between indexed and non-indexed sources, we will sometimes regard classes as families (indexed by themselves via the corresponding identity function). Hence for any object A and any class S of morphisms with domain A, the pair (A,S) will be considered as a source. A particularly useful example is the total source (A,S_A), where S_A is the class of all morphisms with domain A.

If S = (A →^f_i A_i) I is a source and, for each i∈I, S_i = (A_i →^g_ij A_ij) J_i is a source, then the source

(S_i) ◦ S = (A →^{g_ij ◦ f_i} A_ij) i ∈ I, j ∈ J_i

is called the composite of S and the family (S_i)_I.

(1) For a source S = (A → A_i)_I and a morphism f : B → A we use the notation

S ◦ f = ( B →fi ◦f Ai)I .

(2) The composition of morphisms can be regarded as a special case of the composition of sources.

MONO-SOURCES

A source S = (A,f_i)_I is called a mono-source provided that it can be cancelled from the left, i.e., provided that for any pair B →^r ←^s A of morphisms the equation S ◦ r = S ◦ s (i.e., f_i ◦ r = f_i ◦ s for each i ∈ I) implies r = s.

PROPOSITION

(1) Representable functors preserve mono-sources (i.e., if G : A → Set is a representable functor and S is a mono-source in A, then GS is a mono-source in Set).

(2) Faithful functors reflect mono-sources (i.e., if G : A → B is a faithful functor, S = (A,f_i) is a source in A, and GS = (GA,Gf_i) is a mono-source in B, then S is a mono-source in A).

Proof:

(1). If a functor preserves mono-sources, then, clearly, so does every functor that is naturally isomorphic to it. Thus it suffices to show that each mono-source (B →^f_i B_i)_I is sent by each hom-functor hom(A, −) : A → Set into a point-separating source:

(hom(A,B) →^hom(A,f_i) (hom(A,B_i))_I

But this is immediate from the definition of mono-source.

(2). Let G and S be as described. If B →^r ←^s A is a pair of A-morphisms with S ◦ r = S ◦ s, then GS ◦ Gr = G(S ◦ r) = G(S ◦s) = GS ◦ Gs. Since GS is a mono-source, this implies Gr = Gs. Since G is faithful, this gives r = s.

COROLLARY

In a construct (A,U) every point-separating source is a mono-source. The converse holds whenever U is representable.

PROPOSITION

Let T = (S_i) ◦ S be a composite of sources.

(1) If S and all S_i are mono-sources, then so is T .

(2) If T is a mono-source, then so is S.

PROPOSITION

Let (A,f_i)_I be a source.

(1) If (A,f_j)_J is a mono-source for some J ⊆ I, then so is (A,f_i)_I.

(2) If f_j is a monomorphism for some j ∈ I, then (A, f_i)_I is a mono-source.

A mono-source S is called extremal provided that whenever S = S ◦ e for some epimorphism e, then e must be an isomorphism.

PROPOSITION

(1) If a composite source (S_i) ◦ S is an extremal mono-source, then so is S.

(2) If S ◦ f is an extremal mono-source, then f is an extremal monomorphism.

PROPOSITION

Let (A, f_i)_I be a source.

(1) If (A,f_j)_J is an extremal mono-source for some J ⊆ I, then so is (A, f_i)_I.

(2) If fj is an extremal monomorphism for some j ∈ I, then (A, f_i)_I is an extremal mono-source.

The concept of source allows a simple description of coseparators: namely, A is a coseparator if and only if, for any object B, the source (B,hom(B,A)) is a mono-source. This suggests the following definition:

An object A is called an extremal coseparator provided that for any object B the source (B, hom(B, A)) is an extremal mono-source.