3.10. Products of systems
Products of actions
Given a family of actions (αi)i∈I of a category C,
their product is an action β of C defined by ∀CX,
Xβ = ∏i∈I Xαi
∀CX,Y, ∀f∈ Mor(X,Y),
f β = ⨉i∈I f αi
= ⊓i∈I f αi ⚬ πi
∀x∈Xβ, ∀y∈Yβ,
f β(x) = y ⇔ ∀i∈I,
f αi(xi) = yi
Products of families of co-actions are defined in the same way.
The constant product of an action, i.e. on functions from I to each given set,
was already mentioned in 3.6 as the action CI.
Products in categories
A product of a family (Ei)i∈I of objects
in a category C, written C∏i∈I
Ei, is a co-egg (P, π) of the product of co-actions
C(Ei), thus made of an object P with
π ∈ ∏i∈I Mor(P,Ei) such that
all f ↦ (πi∘f)i∈I are bijective :∀CF, ⊓i∈I πi(F)
: P(F) ↔ ∏i∈I
Ei(F)
The products in C of the empty family (I = ∅) are the final objects of C.
The product in the category of sets, coincides with the product of sets (⊓ :
∏i∈I EiF ⥬
PF).
Products in concrete categories
In many concrete categories, any family of objects has a product (P, π) whose role
can usually be played by the product of sets P =
∏i∈I Ei with its natural family of projections
π = ⊓ IdP.
Then in particular, final objets are singletons (even
if other, non-isomorphic singletons may be objects).
Namely, given
any product (P, π), this preferable convention is possible when ⊓π : P
↔ ∏i∈I Ei, transferring the role of P
to ∏i∈I Ei
by this bijection. (Its injectivity already implies for any F,
Inj ⊓i∈I πi(F)).
Our main exceptions will be categories of typed systems: with a set τ of types, a
product of objects with base sets Ei =
∐t∈τ Et,i will have base set
∐t∈τ ∏i∈I
Et,i.
This is identifiable to a subset of ∏i∈I
Ei if I ≠ ∅ but a copy of τ if I = ∅.
On the other hand, if the product as sets P = ∏i∈I
Ei of objects in a concrete category C is otherwise given a role of
object, then the condition for it to serve as the product 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
πi(F) ⊂ ∏i∈I Mor(F,Ei). ∎
By essential uniqueness of products, this also determines all
Mor(P,F) from the category.
If C is a concrete category with products,
(M,e) being an egg can be re-expressed as
∀CE, ∃!f∈Mor(M,EE),
f(e) = IdE.
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 given by the product of sets, with structure 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}
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)
For a symbol s of trajectory
of a tuple in a concrete category,
sP ⊂ ⊓[ ∏i∈I si] with equality if ACI holds.
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 πi(X) 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⚬Lfi =
φi⚬Lπi⚬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))
The structure φP of a constant product P =
EI of an L-algebra E can be written
LP∋(s,x) ↦ sE
⚬ ⊓x
The product of a family of actions of a given monoid, particular case of product of actions
of a category, is also a particular case of product of algebras.
Among systems, aside the case of algebras, any product of partial algebras is a partial algebra ;
any product of injective systems is an injective system ; if ACI holds then
any product over I of serial systems is serial. But the surjectivity of a product system
cannot be ensured unless it is achieved by a single symbol in L.
Set theory and foundations
of mathematics
1. First foundations of
mathematics
2. Set theory
3. Algebra 1
4. Arithmetic and first-order foundations
5. Second-order foundations
6. Foundations of Geometry