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), g•f ∈ Mor(X,X) ∧
1_{X} ∈ Mor(X,X) ∴ g•f = 1_{X}.
Similarly, f•g = 1_{Y}.
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:
 Singletons where relations are constantly true, are final objects in categories
of onetype systems or algebras;
for multitype systems, final objects are made of one singleton per type.
 The empty set where Boolean constants are false is the initial object.
Exercise. Given two fixed sets K and B, consider the category where
 Objects are all (X,φ) where X is a set and φ: X×K→B ;
 Mor((X,φ),(Y,φ') = {f∈Y^{X} 
∀a∈X,∀k∈K, φ(a,k) = φ'(f(a),k)}.
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 subalgebras.
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 ToA be the category where
 Objects are all Cmorphisms f into A, conceived as
(F, f) where F is any object of C, f ∈
Mor_{C}(F,E) and Im f ⊂ A ;
 Mor_{ToA}((F, f),(G, g)) =
{h∈Mor_{C}(F,G)  f = g০h}
The injectivity of f implies the uniqueness of any ToAmorphism to
f.
Now a morphism f in C is an embedding if it is a final object of any
ToA. 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 noninjective 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 subobject),
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 ToA).
But for an image A of noninjective embedding, the new object must stay another set
with a surjection to A.
Products in concrete categories
Let (E_{i})_{i∈I} be a family of objects of
a concrete category C, and its product P =
∏_{i∈I} E_{i}.
For any object F, let X(F) ⊂ P^{F} be the copy of
∏_{i∈I} Mor(F,E_{i}) ⊂
∏_{i∈I} E_{i}^{F} ≅_{F}
P^{F}.
Then, giving P the role of an object of C, this object will be called the
product in C of the E_{i}, written
^{C}∏_{i∈I} E_{i} (such an
object may exist or not depending on C), if
∀F, Mor(F,P) = X(F).
This also determines all
Mor(P,F) from C as we shall see below.
Part of this has an equivalent formulation:
(Inc) ⇔ (∀F, Mor(F,P) ⊂ X(F)) ⇔
(∀i∈I, π_{i}∈Mor(P,E_{i}))
Proof:
Id_{P} ∈ Mor(P,P) ⊂ X(P) ⇒ ∀i∈I, π_{i}∈Mor(P,E_{i})
∀F, ∀f∈Mor(F,P) ∀i∈I, π_{i}∈Mor(P,E_{i}) ⇒ π_{i}০f ∈ Mor(F,E_{i}).∎
Products of algebras
Any data of Lalgebra structures φ_{i} on each E_{i}
defines the one φ_{P} on P, by
(∀i∈I, π_{i}∈Mor_{L}(P,E_{i})) ⇔
(∀i∈I, φ_{i}০^{L}π_{i}
= π_{i}০φ_{P}) ⇔
φ_{P} = ∏_{i∈I} φ_{i}০^{L}π_{i}
This actually defines the product in the category of all Lalgebras F in the sense that ∏_{i∈I} Mor_{L}(F,E_{i}) ≅_{F} Mor_{L}(F,P).
Between Lalgebras E, F,
∀f∈F^{E}, f ∈ Mor_{L}(E,F)
⇔ Gr f ∈ Sub_{L}(E×F).
Proof: ∀s∈L, 
(∀x∈ E^{ns},
f(s_{E}(x)) = s_{F}(f০x))

⇔∀x∈E^{ns},∀y∈F^{ns},(y=f০x⇒
f(s_{E}(x))=s_{F}(y)) 


⇔ ∀(x×y)∈(Gr f)^{ns},
(s_{E}(x), s_{F}(y))∈
Gr f 


⇔ ∀z∈ (Gr f)^{ns},
s_{E×F}(z) ∈ Gr f 
Other proof from previous results:
f∈ Mor_{L}(E,F)
⇔ Id_{E}×f ∈ Mor_{L}(E,E×F)
⇒ Gr f = Im(Id_{E}×f ) ∈ Sub_{L}(E×F).
Gr f ∈ Sub_{L}(E×F)
⇒ π_{0Gr f}∈Mor_{L}(Gr f, E)
⇒ Id_{E}×f = (π_{0Gr f})^{ 1}∈Mor_{L}(E, Gr f)
⊂ Mor_{L}(E, E×F).∎
Products in categories
For any family of objects (E_{i})_{i∈I} of a category C, a data of
an object P with
φ∈∏_{i∈I} Mor(P,E_{i}) is called a product in C,
(P, φ) =
^{C}∏_{i∈I} E_{i}
if it is final in the
category of such data, i.e. all f ↦ (φ_{i}•f)_{i∈I} are bijective :
∀F, ∏_{i∈I}
Hom (F,φ_{i}) :
Mor(F,P) ↔ ∏_{i∈I} Mor(F,E_{i})
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
↪ ∏_{i∈I} E_{i}) ⇒
Inj ∏_{i∈I} Hom (F,φ_{i})
but products as just defined may exist without admitting as such the set theoretical one
(P, π) where P = ∏_{i∈I}
E_{i} and ∏π = Id_{P}.
In particular in categories of typed algebras or other typed systems with a set T of types,
the product of the E_{i} = ∐_{t∈T} E_{t,i}
has underlying set ∐_{t∈T} ∏_{i∈I} E_{t,i}.
This is identifiable to a subset of ∏_{i∈I} E_{i} if I≠∅ but a copy of T if I=∅.
Coproducts
Symetrically by taking the opposite category, a coproduct of a
family of objects (E_{i})_{i∈I} in a category C,
is an initial data (K, j) = ^{C}∐_{i∈I}
E_{i}
of an object K with
j∈∏_{i∈I} Mor(E_{i}, K), i.e. making
bijective all f ↦ (f•j_{i})_{i∈I} :
For all F, ∏_{i∈I}
Hom_{K}(j_{i}, F) :
Mor(K,F) ↔ ∏_{i∈I} Mor(E_{i},F)
In the concrete category of all sets with all functions between them, the coproduct is the
disjoint union with its canonical injections.
But it is usually quite different in other concrete categories (where the j_{i} are still usually injective,
though curiosity exceptions exist).
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. Eggs, basis, clones and varieties
4. Arithmetic and firstorder foundations
5. Secondorder foundations
6. Foundations of Geometry