3.8. Initial and final objects

In any category, an object X is called an initial object if all sets Mor(X,Y) are singletons. Of course any object isomorphic to an initial object is also an initial object, as all isomorphic objects have the same properties. But conversely all initial objects (if they exist) are isomorphic, by a unique isomorphism between any two of them:
For any initial objects X, Y, ∃f∈Mor(X,Y), ∃g∈Mor(Y,X), gf ∈ Mor(X,X) ∧ 1X ∈ Mor(X,X) ∴ gf = 1X.
Similarly, fg = 1Y. Thus f is an isomorphism, unique because Mor(X,Y) is a singleton.∎
By this unique isomorphism, X and Y may be treated as identical to each other. Initial objects are said to be essentially unique.
Similarly, an object X is called a final object if all sets Mor(Y,X)) are singletons.
Such objects exist in many categories, but are not always interesting. For example, in any category of relational systems containing representatives (copies) of all possible ones with a given language: Exercise. Given two fixed sets K and B, consider the category where Does it have an initial object ? a final object ?

Embeddings in concrete categories

In categories of relational systems, any section is an embedding, but without converse in general.
For any subset A of an object of such a category,

(There exists a section with image A) ⇔ (for any object N, Mor(A,N) = {g|A | g∈Mor(E,N)}) ⇒ (any embedding with image A is a section).

For images A of embeddings which are not sections, that formula for Mor(A,N) would not generally fit the concept of morphisms for any relational structure on A. Also in some categories of systems, not all subsets A of systems E are images of embeddings, because they do not fit as objects, particularly if the restricted structure on A fails at some chosen axiom. For example categories of algebras do not accept all subsets as sub-algebras.

Let us generalize the concept of embedding to any concrete category C, making it work the same as in categories of relational systems beyond the case of sections (but unlike sections, it will require checking the whole category). For any subset A of an object E of C, let To-A be the category where

The injectivity of f implies the uniqueness of any To-A-morphism to f.

Now a morphism f in C is an embedding if it is a final object of any To-A. Then it is also a final object of To-(Im f). It is an embedding onto A if moreover A = Im f.
All such embeddings, even non-injective ones, are monic : Section ⇒ Embedding ⇒ Monomorphism.

While C may have been given with pairwise disjoint objects ("forgetting" the canonical injections between them), any image A = Im f of an injective embedding f may receive a role of object added to C (we may call it a sub-object), as an additional representative of an existing isomorphism class in C, by copying to A through f (seen as an isomorphism) the sets of morphisms involving Dom f (independently of the choice of f because of its essential uniquess as a final object of To-A). But for an image A of non-injective embedding, the new object must stay another set with a surjection to A.

Products in concrete categories

For any family of objects (Ei)iI of a concrete category, their product, if it exists, is defined as an object P = ∏iI Ei with sets of morphisms to it defined by

For any object F, Mor(F,P) ≅ ∏iI Mor(F,Ei)

Here, the inclusion (Inc), or rather the canonical injection, of Mor(F,P) into ∏iI Mor(F,Ei) for all F, is equivalent to ∀iI, πi∈Mor(P,Ei).
Any data of L-algebra structures φi on each Ei defines the one φP on P, by

(∀iI, πi∈MorL(P,Ei)) ⇔ (∀iI, φiLi) = πi০φP) ⇔ φP = ∏iI φiLi)

thus (Inc) suffices to determine the algebraic structure of P, and imply the reverse inclusion.
In the more general case, assuming (Inc), the reverse inclusion not only defines Mor(F,P), but also determines all Mor(P,F) from the category if it allows such a product as an object, by making essentially unique any coexisting products of a given family of objects in a given category. This essential uniqueness is verified in the following more general case.

Products and coproducts in categories

A product of any family of objects (Ei)iI in a category, is a final (thus essentially unique) data (P, φ) of an object P with φ∈∏iI Mor(P,Ei), i.e. making bijective all f ↦ (φif)iI :

For all F, ∏iI Hom (Fi) : Mor(F,P) ↔ ∏iI Mor(F,Ei)

Empty products (I=∅) are the final objects. The concept of embedding is a variant of the unary case (itself trivial).

In concrete categories, for any (P, φ) and any F, (∏φ : P ↪ ∏iI Ei) ⇒ Inj ∏iI Hom (Fi) but products as just defined may exist without admitting as such the set theoretical one (P, π) where P = ∏iI Ei and ∏π = IdP.

Symetrically by taking the opposite category, a coproductiI Ei of a family of objects Ei in a category, is an initial data (K, j) of an object K with j∈∏iI Mor(Ei, K), i.e. making bijective all f ↦ (fji)iI :

For all F, ∏iI HomK(ji, F) : Mor(K,F) ↔ ∏iI Mor(Ei,F)

In the concrete category of all sets with all functions between them, the coproduct is the disjoint union with its canonical injections. In useful concrete categories the ji are also usually injective, but curiosity exceptions exist.

Categories of acts

For any concrete category C, let us call egg of C an initial object of the category where Proposition. Any egg (M,e) (if that exists), can be seen in a unique way as a monoid (M,e,•) with an action on every other object X (beyond • on M itself), such that for all objects X, Y, Proof. Defining ∀xX, hx ∈ Mor(M,X) ∧ hx(e) = x, provides an M-structure on each X which interprets each aM in X as defined by the tuple (e,a). So they are preserved: Mor(X,Y) ⊂ MorM(X,Y), which implies the axioms of M-acts.
The composition in M coming as this M-structure for M = X, satisfies the same axioms.
The last axiom of monoid, he = IdM comes from the uniqueness of he obeying its definition, and ensures the reverse inclusion:
g∈MorM(M,X), g = hg(e)g ∈ Mor(M,X). ∎
This monoid (M,e,•) is essentially the opposite of the monoid End(M). Indeed for all a, bM we have ha ∈ End(M), hb ∈ End(M) and hahb(e) = ha(b) = ba.
For any object M of a category C, the concretized category MC has an egg (MM, 1M), with the monoid action directly given by composition. If C was already a concrete category with an egg (M,e) then MC is just a copy of C (with correspondence using the choice of e).

Proposition. For a monoid (M,e, •) seen as an M-set interpreting • as left action, (M, e) is an egg of the category of M-sets; other eggs are the (X,x) where x is a free and generating element of X.

Proof: by properties of acts as algebraic structures and inverses, as x is free and generating in X if and only if (X,x) is isomorphic to (M,e). ∎

Set theory and foundations of mathematics
1. First foundations of mathematics
2. Set theory (continued)
3. Algebra 1
3.1. Morphisms of relational systems and concrete categories
3.2. Algebras
3.3. Special morphisms
3.4. Monoids
3.5. Actions of monoids
3.6. Invertibility and groups
3.7. Categories
3.8. Initial and final objects
3.9. Algebraic terms
3.10. Term algebras
3.11. Integers and recursion
3.12. Presburger Arithmetic
4. Model Theory