3.10. Products of systems
Products in concrete categories
Many usual concrete categories have products (P, π) of any family of objects
(Ei)i∈I, where the base
set of P is usually given by the product of sets P =
∏i∈I Ei with π = ⊓ IdP
(much more often than coproducts have disjoint unions as base sets).
Namely, given
any product (P, π), this preferable convention is possible when ⊓π : P
↔ ∏i∈I Ei, by transferring the role of
this product object from P to ∏i∈I Ei
by this bijection.
(Its injectivity already implies for any F, Inj ⊓i∈I Hom (F,πi)).
Our main exceptions will be categories of typed systems: with a set T of types, a
product of objects with base sets Ei = ∐t∈T Et,i will have base set
∐t∈T ∏i∈I Et,i.
This is identifiable to a subset of ∏i∈I Ei if I≠∅ but a copy of T if I = ∅.
Now, for any family of objects (Ei)i∈I in any
concrete category C, consider the set
P = ∏i∈I Ei, with supposedly given
structures, or anyhow given the role of an object of C (i.e. with precise sets of
morphisms to and from it). Then, the general condition for this object P to play the role of
product of the Ei in C, is that
∀F, ⊓[Mor(F,P)] = ∏i∈I Mor(F,Ei).
The inclusion side of this equality is equivalent to (∀i∈I, πi ∈
Mor(P,Ei)). Indeed,
(πi)i∈I =
⊓ IdP ∈ ⊓[Mor(P,P)] ⊂ ∏i∈I
Mor(P,Ei) ⇒ ∀i∈I, πi ∈
Mor(P,Ei).
Conversely (simple reason) : ∀i∈I, πi ∈ Mor(P,Ei)
⇒ ∀f∈Mor(F,P), πi⚬f ∈ Mor(F,Ei).
Other presentation : ∀F, ⊓[Mor(F,P)] = Im ⊓i∈I
Hom (F,πi) ⊂ ∏i∈I Mor(F,Ei). ∎
Thanks to the essential
uniqueness of products in a given category C, this also determines from C all
Mor(P,F).
Products of relational systems
In a category of all relational L-systems (with fixed L),
any family of systems (Ei, Ei) has a product
(P, P) where P is the product of sets and P =
∩i∈I
Lπi*(Ei)
= ∐s∈L ⊓[ ∏i∈I
si]
For a binary product of L-systems G = E×F these structures
aresG = {x×y | x∈sE
∧ y∈sF} = {z∈Gns
| π0⚬z ∈sE ∧ π1⚬z ∈sF}
where x×y = ((x0,y0),...,
(xns-1, yns-1)).
To check that it forms a product in this category, for any L-system F and any
f = ⊓i∈I fi ∈ PF, i.e.
∀i∈I, fi = πi⚬f ∈ EiF,
f ∈ MorL(F,P) ⇔
Lf[F] ⊂ P
⇔ ∀i∈I, Lf[F] ⊂ Lπi*(Ei)
⇔ ∀i∈I, fi
∈ MorL(F,Ei)
⇔ ⊓f ∈
∏i∈I
MorL(F,Ei)
Products of modules
For any morphism b∈Mor(X,Y) in any category, any product of
b-modules is also a b-module.
Proof. Let (P, π) a product of a family
(Ei)i∈I of b-modules. Then
∀f∈Mor(X,P),
(∀i∈I, ∃!g∈Mor(Y,
Ei), g•b = πi•f)
∴ ∃!h∈Mor(Y,P), ∀i∈I, πi•h•b
= πi•f.
By Inj ⊓i∈I Hom (X,πi) we conclude ∃!h∈Mor(Y,P), h•b = f.
∎
In any category with products, the condition for an object M to be a b-module
can be re-expressed as
∃!h∈Mor(Y,
∏Mor(X,M) M), ∀u∈Mor(X,M),
πu•h•b = u.
If moreover the category is concrete, this condition on h is equivalently written
∀u∈Mor(X,M), ∀x∈X,
πu(h(b(x))) = u(x)
⇔ ∀x∈X, h(b(x)) = πx|Mor(X,M)
Products of algebras
With an algebraic language L, the product P =
∏i∈I Ei of
a family of L-algebras (Ei, φi)
has L-algebra structure φP defined by
(∀i∈I, πi ∈
MorL(P,Ei)) ⇔
(∀i∈I, φi⚬Lπi
= πi⚬φP) ⇔ φP =
⊓i∈I φi⚬Lπi
Indeed, for any L-system F and any
f = ⊓i∈I fi ∈ PF,
f ∈ MorL(F,P) ⇔
φP⚬Lf = f⚬φF
⇔ ∀i∈I, φi⚬Lf =
fi⚬φF
⇔ ∀i∈I, fi
∈ MorL(F,Ei)
⇔ ⊓f ∈
∏i∈I
MorL(F,Ei)
This comes as a particular case
of product of relational systems (as algebras
can be seen as modules):
∀(x,y)∈LP×P, y =
φP(x)
⇔ ∀i∈I, yi = φi
(Lπi(x)).
Morphisms as subalgebras
For any L-algebras E, F,
∀f∈FE, f ∈ MorL(E,F)
⇔ Gr f ∈ SubL(E×F).
Proof: ∀s∈L,
(∀x∈Ens,
f(sE(x)) = sF(f⚬x))
|
⇔ ∀x∈Ens,
∀y∈Fns, (y = f⚬x ⇒
f(sE(x)) = sF(y)) |
|
⇔ ∀(x×y)∈(Gr f)ns,
(sE(x), sF(y)) ∈ Gr f |
|
⇔ ∀z∈(Gr f)ns,
sE×F(z) ∈ Gr f ∎ |
Other proof from previous results:
f ∈ MorL(E,F)
⇔ IdE×f ∈ MorL(E,E×F)
⇒ Gr f = Im(IdE×f ) ∈ SubL(E×F).
Gr f ∈ SubL(E×F)
⇒ π0|Gr f ∈ MorL(Gr f, E)
⇒ IdE×f = (π0|Gr f)-1 ∈ MorL(E, Gr f)
⊂ MorL(E, E×F).∎
Hence another proof of the stability
of equality subsets:
∀f,g∈MorL(E,F),
(Gr f ⋂ Gr g) ∈ SubL (E×F)
∴ {x∈E| f(x) = g(x)} = π0
[Gr f ⋂ Gr g] ∈ SubL E.
Set theory and foundations
of mathematics
1. First foundations of
mathematics
2. Set theory
3. Algebra 1
3.1. Galois
connection
3.2. Relational systems and concrete categories
3.3. Algebras
3.4. Special morphisms
3.5. Monoids
3.6. Actions of monoids
3.7. Invertibility and groups
3.8. Categories
3.9. Initial and final objects
3.10. Products of systems
3.11.
Eggs, basis and varieties
4. Arithmetic and first-order foundations
5. Second-order foundations
6. Foundations of Geometry