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, 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 are

sG = {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), 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:

Hence another proof of the stability of equality subsets:

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