Products and coproducts in the category S ( B ) of Segal topological algebras Mart

Let B be a topological algebra and S(B) the category of Segal topological algebras. In the present paper we show that all coproducts of two objects of the category S(B) always exist. We also find necesssary and sufficient conditions under which the products of two objects of the category S(B) exist.


INTRODUCTION
The study of Segal topological algebras started in [1].It was followed by [2], where the category S(B) of Segal topological algebras was represented as triples (A, f , B) where B was fixed.Further study of the category S(B) was carried out in [3].
The present paper deals with the question of the existence of products and coproducts of objects in the category S(B).While the coproducts exist always and have a form similar to the form of coproducts in the category of algebras, the products might or might not exist and have a bit different description, similar to the description of a Whitney sum known in the theory of fibre spaces.
Let us start by recalling the necessary definitions from [1] and [2].
A topological algebra is a topological linear space over the field K (where K stands for either R or C), in which there is defined a separately continuous associative multiplication.
A topological algebra (A, τ A ) is a left (right or two-sided) Segal topological algebra in a topological algebra (B, τ B ) via an algebra homomorphism f : (2) f is continuous; (3) f (A) is a left (respectively, right or two-sided) ideal of B.
Notice that condition (2) is equivalent to the following condition: used in [1].In what follows, we will denote a Segal topological algebra shortly by a triple (A, f , B).
From now on, we will fix a topological algebra (B, τ B ), which we will not change for this paper.The set Ob(S(B)) of objects of the category S(B) will consist of all Segal topological algebras in the same topological algebra B, i.e., all Segal algebras in the form of triples (A, f , B), (C, g, B), ....
The set Mor((A, f , B), (C, g, B)) of morphisms between Segal topological algebras (A, f , B) and (C, g, B) will consist of all continuous algebra homomorphisms α : A → C with the property g(α(a)) = f (a) for every a ∈ A In [2], it was shown that S(B) is really a category, but not an additive category.In what follows, we will denote by 1 A : A → A the identity map on A for every algebra A, i.e., 1 A (a) = a for every a ∈ A. It is easy to see that if B 0 is a dense left (right or two-sided) ideal of B, then For conciseness of the text, we will write everywhere just "ideal" instead of "left (right or two-sided) ideal".In what follows, every claim about "ideal" holds in all three cases.One just has to fix the sideness of all ideals and then to continue with the same sideness throughout the paper.

FREE PRODUCT OF TWO OBJECTS OF THE CATEGORY S(B)
In algebra it is known that, for any finite collection V 1 , . . .,V n of linear spaces, their tensor product V 1 ⊗ • • • ⊗V n is a linear space and consists of all finite sums of the form where k ∈ N is finite and v j,i ∈ V i for every i ∈ {1, . . ., n}.
It is also known in algebra that, for any collection (A i ) i∈N of linear spaces, their direct sum consists of all tuples (a i ) i∈N with a i ∈ A i for every i ∈ N and a i = θ A i for all but finitely many i ∈ N. Hence, we can write a general element for some k ∈ Z + = {1, 2, . . .}, where there exist j 1 , . . ., j k ∈ N with 1 j 1 < j 2 < j k such that For simplicity, let us denote the element (θ for some k ∈ Z + , some j 1 , . . ., j k ∈ N with 1 j 1 < j 2 < j k , and some b Next, we follow the ideas of [5], p. 9, about the free product of modules over a commutative unital ring.In our case, we will apply them to algebras and give the formulas for algebraic operations for the general element of the free product of two algebras over the field K. Let A and C be algebras, which are made disjoint by setting a = (a, 1) and c = (c, 2) for every a ∈ A and c ∈ C if A ∩C ̸ = / 0 originally.Consider the set By the aforementioned formulas, we can write every element of T in the form for some n, k 1 , . . ., k n , N 1 , . . ., N n ∈ Z + and for some t i, j,l ∈ A ∪C.For every and λ ∈ K, define the operations on T as follows: where and Then T becomes an algebra with respect to those operations.This algebra is called the tensor algebra of A and C. Let for every r ∈ {1, . . ., n}.As the addition of tensor products in T is defined through direct sum, Similarly, as the multiplication of tensor products in T is defined through tensor multiplication, we have that when the elements t i, j,l are considered as elements of the direct summand A of T or of the direct summand ), let T be the tensor algebra of A and C and define a map h T : T → B as follows: for every element Let τ h T be the topology, induced on T by the map h T , i.e., τ h T = {h −1 T (U) : U ∈ τ B }, where τ B denotes the topology of B. Then h T is a continuous map in the topology τ h T .
Notice that h T (s , and h T (λt) = λ h T (t) for every s,t ∈ T and λ ∈ K. Thus, h T is an algebra homomorphism and h T (T ) is closed with respect to the algebraic operations.
Next, let us show that the addition and scalar multiplication are continuous and multiplication is separately continuous in the topology τ h T .For this, let O be an arbitrary neighbourhood of zero in T , t be an arbitrary element of T , and λ an arbitrary scalar from K. Then h T (t) ∈ B and there exists a neighbourhood Since the addition and scalar multiplication are continuous in B and the multiplication is separately continuous in B, there exist neighbourhoods U,V , and Notice that h T (T ) is an ideal of B. Its "sideness" is the same as it is for the dense ideals f (A) and g(C).
Suppose that t i, j,1 ∈ A and b ∈ B. Then hT (t i, j,1 ) ∈ f (A) and there exists ti, j,1 ∈ A such that b hT (t i, j,1 ) = f ( ti, j,1 ) = hT ( ti, j,1 ). ) Similarly, if f (A) and g(C) are right ideals of B, we find elements ti, j,N i in A or C such that Thus, h T (T ) is an ideal of B, which has the same sideness as the ideals f (A) and g(C) had.Moreover, since f (A) ⊂ h T (T ) and f (A) was dense in B, h T (T ) is a dense left (right or two-sided) ideal of B. With this, we have proved the following result.Let I be the two-sided ideal of T , generated by the set for every a 1 , a 2 ∈ A and c 1 , c 2 ∈ C. Let A ⊔C = T /I, equipped with the quotient topology τ A⊔C (induced by the topology τ h T ).Then (A ⊔C, τ A⊔C ) is a topological algebra.
In algebra, the set A ⊔C is called the free product of A and C. Let κ I : T → T /I = A ⊔C be the quotient map and define a map h : With that, we have proved another result.Lemma 2.2.Let (A, f , B), (C, g, B) ∈ Ob(S(B)) and let T be the tensor algebra of A and C. Define the map h T : T → B as in (3) and equip T with the topology τ h T .Let I be the two-sided ideal of T , generated by the set and A ⊔ C = T /I be equipped with the quotient topology.Let κ I : T → T /I be the quotient map.Then the triple (A ⊔ C, h, B), where h(κ I (t)) = h T (t) for every t ∈ T and every κ I (t) ∈ A ⊔ C, is an object of the category S(B).
Take any element y of A ⊔C. Then there exists an element of T such that y = κ I (t).Now, as ν is an algebra homomorphism, Notice that for every i ∈ {1, . . ., n}, j ∈ {1, . . ., k i }, and l ∈ {1, . . ., N i }.Therefore, It is also easy to check that (θ • α)(a) = γ(a) for every a ∈ A and that (θ Take any y ∈ A ⊔C. Then there exists an element of T such that y = κ I (t).By using first (2) and then (1), we obtain that )) .

PRODUCTS IN S(B)
Let us recall from [6] (see Definition in Let us remind that, when A and C are topological algebras, then A ×C = {(a, c) : a ∈ A, c ∈ C}, equipped with the product topology, is also a topological algebra with respect to the algebraic operations defined by for all (a 1 , c 1 ), (a 2 , c 2 ) ∈ A ×C and every λ ∈ K.
In the case of the category of modules over a fixed ring, the product of objects A and C was defined to be A ⊓C = A ×C and the maps α = pr A : A ×C → A and β = pr C : A ×C → C were chosen as projections.
In the category S(B), the conditions α ∈ Mor((A⊓C, h, B), (A, f , B)) and β ∈ Mor((A⊓C, h, B), (C, g, B)) and β =pr C , we would have the condition f (a) = h((a, c)) = g(c), which is not true for all (a, c) ∈ A ×C, in general.Thus, we have to limit ourselves to some subset The construction of D is similar to the construction of the Whitney sum, known for fibre bundles.The difference in our case is that, unlike the case of the Whitney sums of fibre bundles, not all elements b = f (a) of the image f (A) have to have such c ∈ C that g(c) = b, because we do not demand that f (A) = g(C).
Fortunately, D is still an algebra and, choosing the subspace topology on D, induced by the product topology of A × C, we still obtain a topological algebra and are able to define h : D → B by h((a, c)) = f (a) = g(c).But now we can not guarantee that h(D) is dense in B. We faced a similar situation (with a bit more difficult obstacles) in [2], while we were describing the equalizers in the category S(B).
Let us continue with a result similar to Lemma 2 from [2].Remark 4.3.When the present paper had been submitted and was waiting for the opinion of the referees, another paper ( [4]) was written, where the situation of products in the category S(B) was studied for an arbitrary collection of objects in S(B) instead of the product of just two objects.Therefore, several results of the present paper become as a special case of more general results and will be given here without proofs.Now we are ready to give a sufficient condition in order to have a product in the category S(B).

CONCLUSIONS
In the present paper, we showed that all coproducts of elements of the category S(B) of Segal topological algebras exist for every topological algebra B. We also found a necessary and sufficient condition for a topological algebra B under which all products of elements of the category S(B) exist.

A i ) i∈N by 1 ⊕ l=1 b l , where b 1 = θ A 1 .
By doing it, we can write every element of the direct sum in the form k

Lemma 2 . 1 .
Let (A, f , B), (C, g, B) ∈ Ob(S(B)) and let T be the tensor algebra of A and C. Define the map h T : T → B as in (3) and equip T with the topology τ h T .Then (T, h T , B) ∈ Ob(S(B)).

Definition 3 . 1 .
Chapter 5.1, p. 214) that the coproduct of the objects A and B of a category C is a triple (A ⊔ B, α, β ), where A ⊔ B is an object in C and α : A → A ⊔ B, β : B → A ⊔ B are morphisms of the category C such that for every object X in C and every pair of morphisms f : A → X, g : B → X of C there exists a unique morphism θ : A ⊔ B → X of C such that θ • α = f and θ • β = g.Now we will formulate this definition for the category S(B).The coproduct of (A, f , B), (C, g, B) ∈ Ob(S(B)) is a triple ((A ⊔ C, h, B), α, β ), where (A ⊔ C, h, B) ∈ Ob(S(B)), α ∈ Mor((A, f , B), (A ⊔ C, h, B)), β ∈ Mor((C, g, B), (A ⊔ C, h, B)) such that for every (X, j, B) ∈ Ob(S(B)) and every pair of morphisms γ ∈ Mor((A, f , B), (X, j, B)) and δ ∈ Mor((C, g, B), (X, j, B)) there exists a unique morphism θ ∈ Mor((A ⊔ C, h, B), (X, j, B)) such that θ • α = γ and θ • β = δ With this, we are ready to describe the coproducts in the category S(B).Proposition 3.2.For any (A, f , B), (C, g, B) ∈ Ob(S(B)), their coproduct in S(B) exists and is the triple ((A ⊔ C, h, B), α, β ), where (A ⊔ C, h, B) is the object of S(B) described in Lemma 2.2, α : A → A ⊔ C and β : C → A ⊔ C are morphisms defined by α(a) = κ I (a), β (c) = κ I (c) for all a ∈ A, and c ∈ C, where κ I is the quotient map defined in Lemma 2.2.Proof.Let T be the tensor algebra of algebras A and C. Let i A : A → T and i C : C → T be the inclusion maps sending elements of A and B into the direct summands A and C of T , respectively, i.e., i A (a) = a ∈ A ⊂ T and i C (c) = c ∈ C ⊂ T for every a ∈ A and c ∈ C. Then the maps i A and i C are continuous algebra homomorphisms.Moreover, the quotient map κ I : T → A ⊔ C is a continuous algebra homomorphism.Hence, the maps α = κ I • i A and β = κ I • i C are also continuous algebra homomorphisms.Notice that f (a) = h T (a) = h T (i A (a)) and g(c) = h T (c) = h T (i C (c)) for all a ∈ A and c ∈ C. Thus, f = h T • i A and g = h T • i C .By Lemma 2.2, h • κ I = h T .Take any a ∈ A and c ∈ C. Then

Definition 4 . 1 .
Chapter 5.1, p. 217) that the product of the objects A and B of a category C is a triple (A ⊓ B, p, q), where A ⊓ B is an object in C and p : A ⊓ B → A, q : A ⊓ B → B are morphisms of the category C such that for every object X in C and every pair of morphisms f : X → A, g : X → C of C there exists a unique morphism θ : X → A ⊓ B of C such that p • θ = f and q • θ = g.Now we will formulate this definition for the category S(B).The product of (A, f , B), (C, g, B) ∈ Ob(S(B)) is a triple ((A ⊓ C, h, B), α, β ), where (A ⊓ C, h, B) ∈ Ob(S(B)), α ∈ Mor((A ⊓ C, h, B), (A, f , B)), β ∈ Mor((A ⊓ C, h, B), (C, g, B)) such that for every (X, j, B) ∈ Ob(S(B)) and every pair of morphisms γ ∈ Mor((X, j, B), (A, f , B)), δ ∈ Mor((X, j, B), (C, g, B)) there exists a unique morphism θ ∈ Mor((X, j, B), (A ⊓ C, h, B)) such that α • θ = γ and β • θ = δ

Lemma 4 . 2 .
Let (A, f , B), (C, g, B) ∈ Ob(S(B)), D = {(a, c) ∈ A ×C : f (a) = g(c)}, and h : D → B be defined by h((a, c)) = f (a) = g(c) for every (a, c) ∈ D. Consider on A × C the product topology induced by the topologies of A and C and consider on D the subspace topology τ D induced by the product topology on A × C. If D is a subalgebra of D, equipped with the subspace topology, such that h( D) is a dense ideal of B, then ( D, h| D, B) ∈ Ob(S(B)), pr A | D∈ Mor(( D, h| D, B), (A, f , B)), and pr C | D∈ Mor(( D, h| D, B), (C, g, B)).Proof.By the definition of D, conditions (1) and (3) of the Segal topological algebra are fulfilled.It is easy to see, by the definition of h, that h| D= f • pr A | D= g • pr C | D .As f , g, pr A , pr C are all continuous algebra homomorphisms, pr A | D, pr C | D, and h| D are also continuous algebra homomorphisms.Thus, condition (2) of the Segal topological algebra is fulfilled.Hence, ( D, h| D, B) ∈ Ob(S(B)).From the first part of the proof, we also conclude that pr A | D∈ Mor(( D, h| D, B), (A, f , B)) and pr C | D∈ Mor(( D, h| D, B), (C, g, B)).