Category Theoretic (Mono-)/Sources

associativecomposition

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

(1) Whenever convenient we use more concise notations, such as (A, fi)I, (A, fi) or fi

(A →fi Ai)I.

(2)  The indexing class I of a source (A,fi)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 (fi)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, . . . , fn)). 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 (fi)I is a family, i.e., an indexed collection (e.g., the property of being a product). There are other properties of sources (A,fi), depending on the domain A and the associated class {fi|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,SA), where SA is the class of all morphisms with domain A.

If S = (A →fi Ai) I is a source and, for each i∈I, Si = (Aigij Aij) Ji is a source, then the source

(Si) ◦ S = (A →gij ◦ fi Aij) i ∈ I, j ∈ Ji

is called the composite of S and the family (Si)I.

(1) For a source S = (A → Ai)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,fi)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., fi ◦ r = fi ◦ 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,fi) is a source in A, and GS = (GA,Gfi) 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 →fi Bi)I is sent by each hom-functor hom(A, −) : A → Set into a point-separating source:

(hom(A,B) →hom(A,fi) (hom(A,Bi))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 = (Si) ◦ S be a composite of sources.

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

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

PROPOSITION

Let (A,fi)I be a source.

(1) If (A,fj)J is a mono-source for some J ⊆ I, then so is (A,fi)I.

(2) If fj is a monomorphism for some j ∈ I, then (A, fi)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 (Si) ◦ 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, fi)I be a source.

(1) If (A,fj)J is an extremal mono-source for some J ⊆ I, then so is (A, fi)I.

(2) If fj is an extremal monomorphism for some j ∈ I, then (A, fi)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.

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