## 3.10. Products of systems

### Products of actions

Given a family of actions (α_{i})_{i∈I} of a category *C*,
their *product* is an action β of *C* defined by ∀_{C}*X*,
*X*^{β} = ∏_{i∈I} *X*^{αi}

∀_{C}*X*,*Y*, ∀*f*∈ Mor(*X*,*Y*),
*f*^{ β} = ⨉_{i∈I} *f*^{ αi}
= ⊓_{i∈I} *f*^{ αi} ⚬ π_{i}

∀*x*∈*X*^{β}, ∀*y*∈*Y*^{β},
*f*^{ β}(*x*) = *y* ⇔ ∀*i*∈*I*,
*f*^{ αi}(*x*_{i}) = *y*_{i}

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 *C*^{I}.
### Products in categories

A *product* of a family (*E*_{i})_{i∈I} of objects
in a category *C*, written ^{C}∏_{i∈I}
*E*_{i}, 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*,*E*_{i}) such that
all *f* ↦ (π_{i}∘*f*)_{i∈I} are bijective :∀_{C}*F*, ⊓_{i∈I} π_{i}^{(F)}
: *P*^{(F)} ↔ ∏_{i∈I}
*E*_{i}^{(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} *E*_{i}^{F} ⥬
*P*^{F}).

### 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} *E*_{i} with its natural family of projections
π = ⊓ Id_{P}.

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} *E*_{i}, transferring the role of *P*
to ∏_{i∈I} *E*_{i}
by this bijection. (Its injectivity already implies for any *F*,
Inj ⊓_{i∈I} π_{i}^{(F)}).

Our main exceptions will be categories of typed systems: with a set τ of types, a
product of objects with base sets *E*_{i} =
∐_{t∈τ} *E*_{t,i} will have base set
∐_{t∈τ} ∏_{i∈I}
*E*_{t,i}.

This is identifiable to a subset of ∏_{i∈I}
*E*_{i} if *I* ≠ ∅ but a copy of τ if *I* = ∅.
On the other hand, if the product as sets *P* = ∏_{i∈I}
*E*_{i} 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*,*E*_{i}).

The inclusion side of this equality is equivalent to (∀*i*∈*I*, π_{i} ∈
Mor(*P*,*E*_{i})). Indeed,
(π_{i})_{i∈I} =
⊓ Id_{P} ∈ ⊓[Mor(*P*,*P*)] ⊂ ∏_{i∈I}
Mor(*P*,*E*_{i}) ⇒ ∀*i*∈*I*, π_{i} ∈
Mor(*P*,*E*_{i}).

Conversely (simple reason) : ∀*i*∈*I*, π_{i} ∈ Mor(*P*,*E*_{i})
⇒ ∀*f*∈Mor(*F*,*P*), π_{i}⚬*f* ∈ Mor(*F*,*E*_{i}).

Other presentation : ∀*F*, ⊓[Mor(*F*,*P*)] = Im ⊓_{i∈I}
π_{i}^{(F)} ⊂ ∏_{i∈I} Mor(*F*,*E*_{i}). ∎

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

∀_{C}*E*, ∃!*f*∈Mor(*M*,*E*^{E}),
*f*(*e*) = Id_{E}.

### Products of relational systems

In a category of all relational *L*-systems (with fixed *L*),
any family of systems (*E*_{i}, **E**_{i}) has a product
*P* given by the product of sets, with structure **P** =
∩_{i∈I}
^{L}π_{i}*(**E**_{i})
= ∐_{s∈L} ⊓[ ∏_{i∈I}
*s*_{i}]

For a binary product of *L*-systems *G* = *E*×*F* these structures are*s*_{G} = {*x*⊓*y* | *x*∈*s*_{E}
∧ *y*∈*s*_{F}} = {*z*∈*G*^{ns} |
π_{0}⚬*z *∈*s*_{E} ∧ π_{1}⚬*z* ∈*s*_{F}}

To check that it forms a product in this category, for any *L*-system *F* and any
*f* = ⊓_{i∈I} *f*_{i} ∈ *P*^{F}, i.e.
∀*i*∈*I*, *f*_{i} = π_{i}⚬*f* ∈ *E*_{i}^{F},
*f* ∈ Mor_{L}(*F*,*P*) ⇔
^{L}f[**F**] ⊂ **P** ⇔ ∀*i*∈*I*, ^{L}f[**F**]
⊂ ^{L}π_{i}*(**E**_{i})

⇔ ∀*i*∈*I*, *f*_{i} ∈ Mor_{L}(*F*,*E*_{i})
⇔ ⊓*f* ∈ ∏_{i∈I}
Mor_{L}(*F*,*E*_{i})

For a symbol *s* of trajectory
of a tuple in a concrete category,
*s*_{P} ⊂ ⊓[ ∏_{i∈I} *s*_{i}] with equality if AC_{I} 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
(*E*_{i})_{i∈I} of *b*-modules. Then
∀*f*∈Mor(*X*,*P*),
(∀*i*∈*I*, ∃!*g*∈Mor(*Y*,
*E*_{i}), *g*∘*b* = π_{i}∘*f*)

∴ ∃!*h*∈Mor(*Y*,*P*), ∀*i*∈*I*, π_{i}∘*h*∘*b*
= π_{i}∘*f*.

By Inj ⊓_{i∈I} π_{i}^{(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*),
π_{u}∘*h*∘*b* = *u*.

If moreover the category is concrete, this condition on *h* is equivalently written
∀*u*∈Mor(*X*,*M*), ∀*x*∈*X*,
π_{u}(*h*(*b*(*x*))) = *u*(*x*)

⇔ ∀*x*∈*X*, *h*(*b*(*x*)) = π_{x|Mor(X,M)}

### Products of algebras

With an algebraic language *L*, the product *P* =
∏_{i∈I} *E*_{i} of
a family of *L*-algebras (*E*_{i}, φ_{i})
has *L*-algebra structure φ_{P} defined by

(∀*i*∈*I*, π_{i} ∈
Mor_{L}(*P*,*E*_{i})) ⇔
(∀*i*∈*I*, φ_{i}⚬^{L}π_{i}
= π_{i}⚬φ_{P}) ⇔ φ_{P} =
⊓_{i∈I} φ_{i}⚬^{L}π_{i}

Indeed, for any *L*-system *F* and any
*f* = ⊓_{i∈I} *f*_{i} ∈ *P*^{F},
*f* ∈ Mor_{L}(*F*,*P*) ⇔
φ_{P}⚬^{L}f = *f*⚬φ_{F}
⇔ ∀*i*∈*I*, φ_{i}⚬^{L}f_{i} =
φ_{i}⚬^{L}π_{i}⚬^{L}f =
*f*_{i}⚬φ_{F}

⇔ ∀*i*∈*I*, *f*_{i}
∈ Mor_{L}(*F*,*E*_{i})
⇔ ⊓*f* ∈
∏_{i∈I}
Mor_{L}(*F*,*E*_{i})

This comes as a particular case
of product of relational systems (as algebras
can be seen as modules):
∀(*x*,*y*)∈^{L}P×*P*, *y* =
φ_{P}(*x*) ⇔ ∀*i*∈*I*, *y*_{i} =
φ_{i}(^{L}π_{i}(*x*))

The structure φ_{P} of a constant product *P* =
*E*^{I} of an *L*-algebra *E* can be written
^{L}P∋(*s*,*x*) ↦ *s*_{E}
⚬ ⊓*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 AC_{I} 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