# Tag: mono-source

# 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 →^{fi} 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, (f_{1}, . . . , 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 →^{fi} A_{i}) I is a source and, for each i∈I, S_{i} = (A_{i} →^{gij} A_{ij}) J_{i} is a source, then the source

(S_{i}) ◦ S = (A →^{gij ◦ fi} 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

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 →^{fi} B_{i})_{I} is sent by each hom-functor hom(A, −) : A → Set into a point-separating source:

(hom(A,B) →^{hom(A,fi)} (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.