3.9. Initial and final objects
In any category, an object X is called an initial object if all
Mor(X,Y) are singletons. While isomorphic objects have all the same
properties, here is a small converse : any 2 initial objects are isomorphic, by a unique
isomorphism (so, initial objects form either a single isomorphism class, or none):
For any initial objects X, Y, ∃f∈Mor(X,Y),
∃g∈Mor(Y,X), {g∘f, 1_{X}} ⊂ Mor(X,X)
∴ g∘f = 1_{X}.
Similarly, f∘g = 1_{Y}.
Thus f is an isomorphism, unique as ∃!:Mor(X,Y). ∎
By this unique isomorphism, X and Y may be treated as identical to
each other. Initial objects are said to be essentially unique.
Dually, an object X is called a final object if all Mor(Y,X) are
singletons.
For example, in any category of relational systems with a given language
where every isomorphism class of possible systems has representatives :
 The empty set where Boolean constants are false is the initial system ;
 Final systems (with a single type) are the singletons where all relations are constantly true;
 Final multitype systems are made of one singleton per type.
Exercise. Given two fixed sets A and B, consider the category where
 Objects are all (X,ϕ) where X is a set and ϕ: X×A→B ;
 Mor((X,ϕ),(Y,ϕ') = {f∈Y^{X} 
∀x∈X,∀a∈A, ϕ(x,a) = ϕ'(f(x),a)}.
Does it have an initial object ? a final object ?
Eggs
(This is called universal element
in the literature; I use the name "egg" to fit with that of clone.)
Generalizing the concept of regular
element, let us call egg of an acting category C^{α}, any
(M,e) where M is an object and e ∈ M^{α}, such that
∀_{C}E, (Mor(M,E) ∋ f ↦
f^{α}(e)) : E^{(M)} ↔ E^{α}.
which easily implies that 
C^{(M)} is a copy of C^{α},
with correspondence depending on e.
 (M^{α},e)
is an egg of C^{α}.
Equivalently, an egg is an initial object of the category of elements of C^{α} where
 Objects are all data (E,x) of
an object E of C and x ∈ E^{α}
 Mor((E,x),(F,y)) =
{f∈Mor(E,F)  f^{α}(x) = y}
For any object M of a category C, (M,
1_{M}) is an egg of C^{(M)}.
Dually, a coegg of a coacting category C_{β} is an egg of the
opposite acting category, i.e. a final object of its category of coelements (E,x)
where x ∈ E_{β} and Mor((E,x),(F,y)) =
{f∈Mor(E,F)  f_{β}(y) = x}.
In the literature (wikipedia), a category
of elements of some C^{(M)} is called an undercategory and denoted
M/C, while a category of coelements of some C_{(M)}
is called an overcategory and denoted C/M.
Yoneda's lemma can be expressed in our terminology as follows : if
(M,x) is a (co)egg of a (co)action α of a category C, then (α,x)
is an egg of the action on M of the metacategory of all (co)actions
of C.
Precisely, for any object M, any coaction β of C and any
x∈M_{β}, the unique metamorphism ϕ from
C_{(M)} to C_{β} such that
ϕ_{M}(1_{M}) = x is defined by
∀_{C}E, ∀y∈E_{(M)},
ϕ_{E}(y) = y_{β}(x)
Obviously, the image of ϕ is the trajectory of x in C_{β}.
Exercise. Consider the category of sets and their functions
acting by direct images on the powersets of its objects (Set^{⋆}). Does it have an egg ?
Similarly, consider its coaction on these powersets by preimages (Set_{⋆}). Does it have a coegg ?
Hint : use the numbers of elements of involved finite sets.
Subobjects as subcoactions
As introduced in 3.6, any object E of a category C defines there a coaction
C_{(E)}, which is regular admitting (E, 1_{E})
as coegg; then any f∈Mor(E,F) defines by composition a metamorphism
f^{(C)} from C_{(E)} to
C_{(F)}, which
is injective when f is monic. We can then reexpress the concept of subobject
(E,f) of F as the subcoaction Im f^{(C)} of
C_{(F)} (also known as the trajectory of (E,f) there),
which is metaisomorphic to C_{(E)}.
This makes the concept of subobject strictly independent of a choice of presentation (but
ontologically more expensive). So, a subobject of F can finally be understood
as a regular subcoaction of C_{(F)}, whose coeggs are its
presentations. Indeed from definitions, if (E,f) is a coegg of some subcoaction
of C_{(F)} then f is monic from E to F.
This will let us define a subobject of F by the data of a subcoaction of
C_{(F)}, under the definiteness condition that this subcoaction needs
to be regular. In particular, this can be used to define concepts of direct images and preimages
of subobjects by morphisms, as given by the obviously expressible concepts of direct images and preimages of subcoactions.
Its meaningfulness on subobjects, is precisely the question whether the result from a
regular subcoaction is also regular. This usually holds
for preimages in categories of systems, as can be shown by explicitly describing the result as
a subsystem.
But, the situation for direct images can more often be somewhat subtle,
except for monomorphisms with which it is trivial (a presentation of the result is given by
composition). The question of defining a direct image of a subobject (X,u)
of E by an f∈Mor(E,F), comes down to defining an image of
the morphism f∘u∈Mor(X,F) as a subobject of F
(beyond the trivial case
when f∘u is monic). Of course, like in all such constructions of subobjects,
it is accepted as a subobject when the trajectory of (X,f∘u) in
C_{(F)} is regular. Yet, unlike for many other cases, a concept of
image of a morphism may still be accepted beyond this case, because it comes naturally given
by the explicit descriptions of this image as a subsystem (a simple example can be found in the
category of relational systems with 2 unary relation symbols). Then it fits anyway the following wider
abstract definition of the image of a morphism: the subobject it generates (the smallest of the
subobjects which contain it).
Embeddings in concrete categories
Let us generalize the concepts of embedding and preembedding from
categories of relational systems to any concrete category C.
A morphism f ∈ Mor(E,F) will be called a preembedding
if ∀_{C}X, Mor(X,E) =
{g∈E^{X}  f⚬g ∈ Mor(X,F)}
This formula actually implies f∈Mor(E,F).
In other words, while f∈Mor(E,F) makes (E, Id_{E})
an element of the coaction giving to
each X the set {g∈E^{X}  f⚬g ∈ Mor(X,F)} (subcoaction of C_{E}),
f is a preembedding when (E, Id_{E}) is a generator and thus
also a coegg of this coaction.
Then, an embedding is an injective preembedding, i.e. an
f : E ↪ F such that, equivalently
∀_{C}X, Mor(X,E) =
{f^{ 1}⚬h  h ∈ Mor(X,F) ∧
Im h ⊂ Im f}
∀_{C}X,
{h ∈ Mor(X,F) 
Im h ⊂ Im f} = {f⚬g  g∈Mor(X,E)}
Let us introduce a related concept.
Any fixed subset A ⊂ F defines a subcoaction
C_{(A)} of C_{(F)} by
∀_{C}X, X_{(A)} =
{g∈X_{(F)}  Im g ⊂ A}
Now let us call quasiembedding any f∈Mor(E,F) such that
(E,f) is a coegg of some C_{(A)} and thus also of C_{(Im f)} :
∀_{C}X, ∀g∈Mor(X,F), Im g ⊂ Im f ⇒
∃!ϕ∈Mor(X,E), f ⚬ ϕ = g
(Inj f implies one side of this condition: ∀g∈Mor(X,F),
!ϕ∈Mor(X,E), f ⚬ ϕ = g)
In most
useful concrete categories, all quasiembeddings will be
embeddings ; exceptions are easy to build in other categories designed for this purpose.
Dependencies between some properties of morphisms
The diverse properties of a morphism f∈Mor(E,F) are related as follows.
 If Im f ⊂ A ⊂ F then
((E,f) is coegg of C_{(A)}) ⇔ (f is a quasiembedding and
∀g∈Mor(E,F), Im g ⊂ A ⇒ Im g ⊂ Im f)
 Injection ⇒ (preembedding ⇔ quasiembedding)
 Quasiembedding ⇒ monomorphism
 (Monomorphism ∧ preembedding) ⇒ injection
 Section ⇒ embedding
Proofs. ∀x∈E, ϕ = (E∋y ↦ (f(y) = f(x) ?
x : y)) ⇒ f ⚬ ϕ = f ⇒ (ϕ ∈ End E ∴ ϕ = Id_{E}).

∃h∈Mor(F,E), h ⚬ f = 1_{E} ∴
∀g∈E^{X}, f ⚬ g ∈ Mor(X,F) ⇒
g = h ⚬ f ⚬ g ∈ Mor(X,E).∎
Let f∈Mor(E,F) a quasiembedding.
If there exists a preembedding g∈Mor(X,F) with the same image Im g = Im
f = A ⊂ F and
AC_{A} holds then f is an embedding.
Proof.
∃h∈X^{A}, g⚬h = Id_{A} ∴
g⚬h⚬f = f ∈ Mor(E,F) ∴ h⚬f
∈ Mor(E,X)
∃ϕ∈Mor(X,E), f⚬ϕ = g ∴ f⚬ϕ⚬h⚬f
= g⚬h⚬f = f ∴ ϕ⚬h⚬f = Id_{E}.∎
A subobject (X,u) of E will be qualified as embedded
(resp. quasiembedded) if u is an
embedding (resp. a quasiembedding) from X to E. This does not depend
on the choice of presentation of a given subobject.
If a subset A of an object F is the image of an embedding f∈Mor(E,F), this gives
A the status of an embedded subobject (A, Id_{A}) ≡ (E,f).
(For a quasiembedding we can do similarly with a copy of E attached to A and thought of
as independent of E).
Then for any object X, we can define Mor(X,A) from Mor(X,F)
directly as X_{(A)}, while Mor(A,X) is only directly defined
from Mor(F,X) if f is a section, as {g_{A} 
g∈Mor(F,X)}.
Equalizers
We defined in 3.3 the equalizer
Eq(f, g) ⊂ E of any two functions f,g with the same domain
E, and found this subset of E to be more precisely a subalgebra when
f,g are morphisms in a category of algebras.
Let us generalize this to any category C, conceiving the equalizer Eq(f, g)
of two morphisms f,g∈Mor(E,F) as a subobject of E,
inspired by the above concept of quasiembedding.
Namely, we first define the subcoaction C_{(f=g)} of C_{(E)} by
∀_{C}X, X_{(f=g)} =
{h∈X_{(E)}  f∘h = g∘h}
(which is stable, as an equalizer of metamorphisms for the metaalgebraic structure of coaction).
Then we define the equalizer Eq(f, g) as the subobject of E presented by the
coeggs of this C_{(f=g)} if it is regular.
In any concrete category, equalizers are particular quasiembedded subobjects, namely
the coeggs of the C_{(A)} for subsets A defined as equalizers
of two morphisms seen as functions, since C_{(f=g)}
coincides with C_{(Eq(f, g))}.
Any section is an equalizer. Namely, if f∈Mor(E,F),
g∈Mor(F,E) and g∘f = 1_{E}
then the section f is an equalizer of (1_{F}, f∘g).
Submodules
Let X,Y,F be objects of a concrete category and
b∈Mor(X,Y). A subset A ⊂ F
will be called bstable if
∀g∈Mor(Y,F), Im g⚬b ⊂ A ⇒
Im g ⊂ A.
In particular:  In any category made of Lsystems for some algebraic language L,
if 〈Im b〉_{L} = Y then any Lstable subset is bstable.
 If b is surjective then all subsets of objects are bstable.
Let us call bsubmodule of an object F, any subobject
(E,f) of F such that E is a bmodule.
For any bmodule F and f∈ Mor(E,F),
 If f is a preembedding and b is bijective then E is a bmodule;
 If f is a quasiembedding and Im f is bstable then
(E,f) is a bsubmodule.
Proofs:
 ∀u∈Mor(X,E), f⚬u⚬b^{1} ∈
Mor(Y,F) ∴ u⚬b^{1}∈Mor(Y,E).
 ∀u∈Mor(X,E), (∃!g∈Mor(Y,F),
g⚬b = f⚬u ∴ Im g ⊂ Im f)
∴ (∃!h∈Mor(Y,E), f⚬h⚬b = f⚬u)
∴ (∃!h∈Mor(Y,E), h⚬b = u).∎
This stability concept can be generalized from subsets to subobjects in abstract categories.
To that case, the result 2. is easily extended : any bstable subobject of a bmodule
is also a bmodule. This is of special interest for equalizers, since any equalizer is stable by
any epimorphism. The details are left as an easy exercise for the reader.
Set theory and foundations of mathematics
1. First foundations of mathematics
2. Set theory
3. Algebra 1
4. Arithmetic and firstorder foundations
5. Secondorder foundations
6. Foundations of Geometry
Other languages:
FR : 3.9.
Objets initiaux et finaux