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Серия «Математика»
2023. Т. 44. С. 116—135
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Research article
УДК 512.53, 512.58 MSC 20M30, 18M05
DOI https://doi.org/10.26516/1997-7670.2023.44.116
On Chu Spaces over SS — Act Category
Evgeniy E. Skurikhin1'2, Alena A. Stepanova1^, Andrey G. Sukhonos1
1 Far-Eastern Federal University, Vladivostok, Russian Federation
2 Institute of Applied Mathematics FEB RAS, Vladivostok, Russian Federation И stepltd@mail.ru
Abstract. We prove the general properties of morphisms of Chu spaces and functors with a value in the category Chu(SS — Act) of Chu spaces over the category SS — Act. As a consequence, for the category Chu(SS — Act) the existence of coproducts and some products is proved, monomorphisms and epimorphisms are characterized; in terms of this category the characteristics of separable and complete separable Chu spaces are given.
Keywords: Cartesian closed category, S-Act, Chu spaces, functors, limits
Acknowledgements: The research was carried out at the Far Eastern Center for Mathematical Research with the financial support of the Ministry of Science and Higher Education of the Russian Federation, Agreement No. 075-02-2023-946 dated February 16, 2023 on the implementation of programs for the development of regional scientific and educational mathematical centers.
For citation: SkurikhinE. E., Stepanova A. A., Sukhonos A. G. On Chu Spaces over SS — Act Category. The Bulletin of Irkutsk State University. Series Mathematics, 2023, vol. 44, pp. 116-135.
https://doi.org/10.26516/1997-7670.2023.44.116
Научная статья
0 пространствах Чу над категорией S S — Act Е. Е. Скурихин1,2, A. A. Степанова1^, A. Г. Сухонос1
1 Дальневосточный федеральный университет, Владивосток, Российская Федерация
2 Институт прикладной математики ДВО РАН, Владивосток, Российская Федерация
KI stepltd@mail.ru
Аннотация. Доказываются общие свойства морфизмов пространств Чу и функторов со значением в категории Chu(SS — Act) пространств Чу над категорией S S — Act. В качестве следствий доказывается существование копроизведений, некоторых произведений, характеризуются мономорфизмы и эпиморфизмы категории Chu(SS — Act); в терминах этой категории даются характеристики отделимых и полных отделимых пространств Чу.
Ключевые слова: декартова замкнутая категория, б'-полигоны, пространства Чу, функторы, пределы
Благодарности: Работа выполнена в Дальневосточном центре математических исследований при финансовой поддержке Минобрнауки России, соглашение № 07502-2023-946 от 16 февраля 2023 г. по реализации программ развития региональных научно-образовательных математических центров.
Ссылка для цитирования: SkurikhinE. E., Stepanova A. A., Sukhonos A. G. On Chu
Spaces over S S — Act Category // Известия Иркутского государственного университета. Серия Математика. 2023. Т. 44. C. 116-135. https://doi.org/10.26516/1997-7670.2023.44.116
1. Introduction
A left 5-act, or simply S-act, over monoid S is a set A upon which S acts unitarily on the left. A mapping f : A —> B is called homomorphism of 5-acts if f (sa) = sf (a) for any a e A, s e S, f (sa) = sf (a) [2].
The category whose objects are 5-acts, morphisms are homomorphisms of 5-acts, and the composition of morphisms is defined as a superposition of the corresponding maps, is denoted by S — Act, so that Ob(S — Act) is the class of all 5-acts, Homs-Act(A, B) is the set of all homomorphisms from S-act A to S-act B, the units of the S — Act category are the identical mappings 1A e HomS-Act(A, A).
The monoids action on sets occurs in various situations. General properties of 5-acts are actively studied, as well as classes of 5-acts with specific properties [2;3;7;8]. 5-acts are special cases of presheaves of sets on categories and are therefore related to Grothendieck toposes theory. This
makes it possible to obtain results about S -acts, as special cases of the results about the presheaves [5].
In [9], the category Chu(V) is introduced. Its objects are Chu spaces r : A 0 X —> D, a morphism from r to r' : Ä 0 X' ^ D' is an arbitrary triple (f,g,h) of morphisms f : A ^ N, g : X' ^ X, h : D ^ D' in the category V suchthat h o r o (1^ 0 g) = r' o (f 01x'). In [9-11], this category was studied for the case when V is the S — Act category, S is a commutative monoid and the product is a tensor product. In this paper, we study the category Chu(SS — Act) where SS — Act is a category introduced in [6] and is an extension of the category S — Act.
2. Preliminary Results
Define the category S S - Act as follows [6]: Ob(SS -Act) = Ob(S - Act), Homss-Act(A,B) = Homs-Act(S x A,B), the composition of morphisms u G HomsS-Act(A,B) and v G HomsS-Act(B,C) is defined by equality (u ■ v)(s,a)=v(s,u(s,a)), where s G S, a G A, the identity morphisms in S S — Act are the morphisms eA G Homss-Act(A, A) where ca(s, a) = a.
The Chu space over the S S—Act category is defined as the set (A,X,D,r), where A,X,D G Ob(SS - Act), r G Homss-Act(A x X,D). If this is not confusing, for Chu space (A, X, D, r) we will use the notation r G Homss-Act(A x X,D).
In accordance with the general definitions [1], we define the category Chu(SS - Act). Let r G Homss-Act(A x X,D), r1 G Homss-Act(A x X', D'). A morphism or a Chu transform of r into r' is a triple (f,g, h) of f G Homss-Act(A,A'), g G Homss-Act(X',X), h G Homss-Act(D,D') such that h ■ r ■ (eA x g) = r' ■ (f x ex'). In this case we will write (f,g,h) : r ^ r'. If (f',g',h') : r' ^ r'' then the composition of Chu transforms is defined as follows:
(f, g', h' ) o (f, g, h) = (f' ■ f,g ■ g', h' ■ h) : r ^ r" .
3. Main Lemma
If и : S x A ^ В is a homomorphism of 5-acts, t G S then the mapping tu : S x A ^ В, given by the equality (tu)(s,a) = u(st,a), is also a homomorphism, so that the set Homss-Act(A, B) is endowed with the 5-act structure. Similarly to set mappings, we introduce the notation: Homss-Ac t(A,B) = BA.
In [6] it is proved that the category S S — Act is Cartesian closed, i.e., the functors HomsS-Act(• x •, •) and HomsS-Act(•, 'Hss(•, •)) are isomorphic for some functor Hss : (SS — Act)0 x (SS — Act) ^ SS — Act.
Let us define the functor Hss. If A,B,A',B' e Ob(SS — Act), f e Homss-ActXA',A), g e HomSs-Act(B, B') then
Uss(A, B) = BA = HomSS-Act(A, B)
and the mapping gf = %ss(f,g) e Homss-Act(BÄ, BlA') is defined as follows:
gf(s,w) = nss(f,g)(s,w) = (sg) ■ w ■ (sf),
where (s,w) e S x BA.
Denote by pa,x,d : HomSs-Act(A x X,D) ^ HomSs-Act(A,Dx) a mapping such that ((pa,x,d(0)(s,a))(t,x) = f(ts, (ta,x)), where s,t e S, a e A, x e X, re Homss-Act(A x X, D).
In [6] it is proved that every mapping pa,x,d is bijective and
(*) Pa',x',d' (h ■r ■ (v x g)) = ha ■ pa,x,d (r) ■ v
for all
v e Homss-Act(A',A), g e HomSs-Act(X',X), h e HomSs-Act(D, D'), r e Homss-Act(A x X,D). Thus the family of mappings
Pss = [pa,x,d | A, X,D e Ob(SS - Act)}
is an isomorphism of functors
Pss : Homss-Act(• x •, •) ^ HomSS-Act(•, Hss(•, •)).
The equality (*) is equivalent to
(**)
Pa,x',d'(h^(eAxg)) = hg■pa,x,d(r) and p^,x,D(r■(vxex)) = Pa,x,d(r)^v. For r e Homss-Act(A x X, D), we introduce the notation:
r = Pa,x,d (r).
By vxd e Homss-Act(Dx x X, D) we also denote the Chu space such that
pdx ,x,d (rXD) = fxb = eDx,
where eDx : S x Dx ^ Dx is a unit in the category SS — Act. Note that r e Homss-Act(A,Dx).
Lemma 1. (main Lemma) 1) Let
f e HomSs-Act(A,A'), g e HomSs-Act(X',X), h e Homss-Act(D,D'), r e HomSs-Act(A x X, D), r' e HomSs-Act(A' x X', D'). Then
(a) (f, g, h) e Homchu(ss-Act) (r, r') h9 ■ pa,x,d(r) = PA',X',D' (r') ■ f
^ h9 • r = r' • f;
(b) (h9, g, h) e HomChu{ss-Act)(rxD, rX'D');
(c) (r, ex, eD) e Homchuiss-Act)^ rxD);
(d)
(f,g,h) e Homchu(ss-Act)(r,rx'D') ^ f = h9 • pa,x,d(r) f = h9 • r, in particular,
(e) (f,g, h) = (h9,g, h) о (f,ex,&d).
2) For all w e Homss-Act(A, Dx), we have pa,x,d(rXD • (w x ex)) = w.
3) There is equality
(f) r = rxD • (r x ex)
and for w e Homss-Act(A, Dx), r = rxD • (w x ex), we have w = r.
4) For all w e Homss-Act(A, Dx) the following conditions are equivalent:
(g) PA,x, d (r)= w;
(h) (w, ex, eD) e Homchu(ss-Act)(r, rxD);
(i) r = rxD • (w x ex).
Proof. Let us prove 1). (a) By definition of morphisms of Chu spaces, we have
(f,g, h) e Homchu(ss-Act)(r, r') ^ h • r • (eA x g) = r' • (f x ex'). Since pA,X',D' is bijective then
h^r•(eA xg) = r'•(/xex') ^ Pa,x',d'(h^ieA xg)) = pA,x',D'(r'•(f xex')).
By (**), we have
Pa,x',d' (h • r • (eA x g)) = h9 • p a,x,d (r) = h9 • r, Pa,x',d'(r' • (f x ex')) = Pa',x',d'(r') • f = r' • f.
Hence the desired result is obtained.
(b) Since PxD'Xd(rXD) = &dx and pDx'xX'D(rx'D') = eDx' then by (a) we have
(f,g,h) e Homchu(ss-Act)(rxD,rx'D) if and only if h9 • Pdx )X,d (rXD) = PDx' xD (rx'D') • f if and only if h9 = f.
Известия Иркутского государственного университета. Серия «Математика». 2023. Т. 44. С. 116-135
(c) By (a), we have (f, ex, eD) e HomChu(ss-Act)(r, rXD) if and only if Pa,x,d (r) = pDx ,S,D (rXD) ■ f if and only if r = f.
(d), (e) Since pD,x> x> d' (rx'D') = e^'X' then by (a) we have
(f,g,h) e HomChu(ss-Act) (r, rX'D')
if and only if h9 ■pa,x,d (r) = Pa',x',d' (rX'D' )-f if and only if h9 ■pa,x,d (r) = f. By (b) and ( c), we have
(h9,g,h) e Homchu(ss-Act)(rxD, rX',D'), (r, ex, eD) e Homchu(ss-Act)(r, rXD). Hence (h9,g, h) o ( r, ex, en) = (h9 ■ r,g ■ ex, en ■ h) = (f, g, h).
Let us prove 2). By (**), we have pai,x,d(f ■ (v x ex)) = pa,x,d(f) ■v for all v e HomSs-Act(A',A). If v = w e HomSs-Act(A,Dx) and r = rXD then pa,x,d(rXD ■ (w x ex)) = pDx,X,D( rXD) ■w = w.
Let us prove 3). Let r e Homss-Act(A x X, D). If w = pa,x,d(f) = r in 2) then pa,x,d(rXD ■ (r x ex)) = r = Pa,x,d(r). Since pa,x,d is an injective, then rXD ■ ( r x ex) = r.
If fXD ■ (w x ex) = rXD ■ ( r x ex) for some w e Homss-Act(A, Dx) then by 2) we have w = pa,x,d(rXD ■ (w x ex)) = Pa,x,d(rXD ■ (r x ex)) = r.
Let us prove 4). The equivalence of the conditions (g) and ( i) is proved in 2), and the equivalence of the conditions (g) and (h) is verified by proving (c). □
4. Monomorphisms and epimorphisms in the category
Chu(SS — Act)
Let us give conditions characterizing epimorphisms and monomorphisms in the category Chu(SS — Act).
Theorem 1. Let r e Homss-Act(A x X,D) and r' e Homss-Act(A' x X',D'). Then a morphism (f,g,h) e HomChu^SS-Act)(r,r') is an epimor-phism if and only if f e Homss-Act(A, A') is an epimorphism, g e Homss-Act(X', X) is a monomorphism and h e Homss-Act(D, D') is an epimorphism.
Proof. Necessity. Let (f, g, h) : r ^ r' is an epimorphism in the category Chu(SS — Act).
We will show that f is an epimorphism in the category SS — Act. Let fi, /2 e Homss-Act(A', E) such that f\ ■ f = f2 ■ f. It is necessary to prove the equality f1 = f2. Define
w e HomSS-Act((E x A') x X', D')
and (f[,ex', eD'), (f2, ex',eD') G HomChu(ss-Act)(r', w) as follows:
w(s, ((e, a'),x')) = r'(s, (a1, x')), f[ (s,a' ) = (fi(s,a' ),a! ), f2(s,a') = (Ms, a'),a')
for all s G S, a' G N, x' G X', e G E .It is not difficult to understand that the definition of Chu transforms (f[,eX',eD), (f2,ex,eD>) is well defined and equality (f[,ex',eD') ■ (f,9,h) = (f2,^x',&D') ■ (f,9,h) is true. Since (f,g,h) is an epimorphism in the category Chu(SS - Act) then f[ = f2. Thus, f1 = f2 and f is an epimorphism in the category S S - Act.
Now we will show that g is a monomorphism and h is an epimorphism in the category S S - Act. Let
91,92 G Homss-Act(E,X' ),hi,h2 G Homss-Act(D', F )
such that g ■ g1 = g ■ g2 and h1 ■ h = h2 ■ h. It is necessary to prove the equality g1 = g2 and h1 = h2. Define r2, r2 G Homss-Act((A' x E) x F by equalities r1 = hf1 ■ r', r2 = hg2 ■ r'. By Lemma 1, hg ■ r = r' ■ f. Hence
n ■ f = hf ■ r' ■ f = hf1 ■ h9 ■ r = (h1 ■ h){^gi)r =
= (h2 ■ h)^92)r = hg22 ■ h9 ■ r = hf ■ r' ■ f = r2 ■ f,
that is r1 ■ f=r2 ■ f.
Since f is an epimorphism in the category S S - Act then r1 = r2. Therefore r1 = r2 = r0 : & x E ^ F, r0 : N ^ FE. By Lemma 1(a) the equality r1 = r0, or equivalent equality h1 ■ r' = r0 ■ ca , means that (eA ,g1,h1) : r' ^ r0 is a homomorphism of Chu spaces. Similarly, (ca',g2,h2) : r' ^ r0 is a homomorphism of Chu spaces too. By the definition of composition of Chu spaces morphisms, we have
(eA',91, hi) o (f,g,h) = (e,A> ■ f,9 ■ 91M ■ h) =
= (eA' ■ f,9 ■ 92, h2 ■ h) = (eA,92^2) o (f,g,h).
Since (f,g, h) is an epimorphism in the category S S - Act then
(eA ,91,h1) = (eA ,92M ),
so g1 = g2, h1 = h2. Thus, g is a monomorphism and h is an epimorphism.
Sufficiency follows directly from the definition of the composition of morphisms of Chu spaces. □
Theorem 2. Let r G Homss-Act(A x X,D) and r' G Homss-Act(A' x X', D'). Then a morphism (f, g, h) G Homchu(ss-Act)(r, r') is a monomorphism if and only if f G Homss-Act(A, A1 ) is a monomorphism<,, g G Homss-Act(X',X) is an epimorphism and h G Homss-Act(D,D') is a monomorphism.
Proof. Necessity. Let (f,g,h) e Homchu(ss-Act) (r,r') is a monomor-phism.
We will show that h is a monomorphism in the category SS — Act. Let h1,h2 e Homss-Act(E, D) such that h ■ h1 = h ■ h2. It is necessary to prove the equality h1 = h2. Define w e Homss-Act(A x X,E UD) and (eA, ex, hi), (eA, ex,h2) e HomChu(ss-Act)(w, r) as follows:
w(s, (a, x)) = r(s, (a, x)), hi(s, e) = h1(s, e),h'2(s, e) = h2( s, e), h1(s, d) = h2(s, d) = d
for all s e S, a e A, x e X,e e E,d e D. It is not difficult to understand that the definition of Chu transforms (eA, ex,hi), (eA, ex,h'2) is well defined. Since h■ h1 = h■ h2 then ( f,g, h) ■ (eA, ex, hi) = ( f,g, h) ■ (eA, ex, h'2). Since ( f,g,h) is a monomorphism in the category Chu(SS — Act) then hi = h'2. Thus, hi = h2.
Now we will show that g is an epimorphism in the category SS — Act. By Lemma 2 [11], it enough to show that the morphism g : S x X' ^ S x X is epimorphism in the category S — Act, where g(s, x') = (s, g(s, x')) for s e S, x' e X'. Assume the converse, i.e., Xi = S xX, where Xi = g(S x X'). By X0 we denote the Rees factor act of 5-act S x X by the Rees congruence pxx. Define
w e HomSS-Act(A x (Xo xX),D)
and
(eA, gi, eD), (eA, g2, eD) e HomChu(ss-Act)(w, r)
as follows:
w(s, (a, (x0,x))) = r(s, (a, x)), gi(s ,x) = (Xi,x), §2(s ,x) = ((s ,x)/pXl ,x),
for all s e S, a e A, x e X, x0 e X0. Obviously gi = g2. From the definition of the Chu space w follow the definitions of the Chu transforms (eA, gi, eD), (eA, g2, en) are well defined. It is not difficult to understand that gi ■g = g2 ■ g. Hence (f,g,h) ■ (eA, gi, eD) = (f,g,h) ■ (eA, g2, eD). Since (f,g,h) is a monomorphism in the category Chu(SS — Act) then gi = g2, contradiction. Thus, g is an epimorphism in the category SS — Act.
Finaly we will show that f is a monomorphism in the category SS — Act. Let fi, f2 e Homss-Act(E, A) such that f ■ fi = f ■ f2. It is necessary to prove the equality fi = f2. Define
n, r2 e HomSs-Act(E x X', D)
as follows:
ri(s, (e, x')) = r(s, (fi(s,e),g(s,x'))), r2(s, (e, x')) = r(s, (M.s, e),g(s, x')))
for all x' £ X', s £ S, e £ E. Since (f,g,h) £ Homchu(ss-Act)(r,r') then
(h ■ n)(s, (e,x')) = h(s,n(s, (e,x'))) = h(s,r(s, (fi(s,e),g(s,x')))) = = r'(s, (f (s, fl(s, e)), x')) = r'((f ■ fl)(s, e),x'))
for all x' £ X', s £ S, e £ E. Similarly,
(h ■ r2)(s, (e, x')) = r'(s, ((f ■ f2)(s, e)), x')
for all x' £ X', s £ S, e £ E. Since f ■ f1 = f ■ f2 then
(h ■ n)(s, (e, x')) = (h ■ r2)(s, (e, x')),
i.e., h ■ r1 = h ■ r2. A morphism h is a monomorphism in the category SS — Act. Hence n = r2, i.e., r(s, (fi(s,e),g(s,x'))) = r(s, (f2(s,e),g(s,x'))) for all x' £ X', s £ S, e £ E. Let x £ X, s £ S, e £ E.
We will prove the equality r(s, (f1(s,e),x)) = r(s, (f2(s,e),x)). Since g is an epimorphism in the category SS — Act, then by Lemma 2 [11], gs is surjective and x = g(s,x') for some x' £ X'. Then r(s, (f1(s,e),x)) = r(s, (Ms,e),g(s,x'))) = r(s, (f2(s,e),g(s,x'))) = r(s, (f2(s,e),x)). Define w £ HomsS-Act(E x X,D) as follows: w(s, (e,x)) = r(s, (f1(s,e),x)). Since
r(s, (Ms, e), x))= r(s, (M.s, e), x)) = w(s, (e, x)) = eD(s, w(s, (e, ex(s, x))))
then (/1,ex,eD), (¡2,ex) £ HomChu(ss-Act)(w,r). Obviously,
(f,9,h) ■ (f1 ,ex,e-D) = (f,g,h) ■ (h,ex,eD).
Since (f,g,h) is a monomorphism in the category Chu(SS — Act) then h = f2.
Sufficiency follows directly from the definition of the composition of morphisms of Chu spaces. □
5. Separable Chu space
The Chu space r e Homss-Act(A x X,D) is is called separable (complete separable) if r = pa,x,d(0 e Homss-Act(A, Dx) is a monomorphism (isomorphism) in the category SS — Act.
Proposition 1. (on separable and complete separable Chu spaces)
1) For Chu space r G Homss-Act(A x X,D), the following conditions are equivalent:
(a) r is separable;
(b) (r, ex, e-D) is a monomorphism in the category Chu(SS-Act), where (r, ex, eD) G Homchu(ss-Act)(r, rXD);
(b') there exists a monomorphism w G Homss-Act(A,Dx) in the category SS — Act such that (w, ex, eD) G HomChu(ss-Act)(r, rxD);
(c) there exists a morphism (f,g,h) G HomChu(SS-Act)(r, rx,D) such that f is a monomorphism in the category S S — Ac t.
2) Let (f,g,h) G HomChu^SS-Act)(r,r'). If f is monomorphism in the category S S — Act and r' is a separable Chu space then r is a separable Chu space.
3) For Chu space r G Homss-Act(A x X,D), the following conditions are equivalent:
(d) is complete separable;
(e) (r, ex, ев) is an isomorphism in the category Chu(SS — Act), where (r, ex, eD) G Homchu(ss-Act)(r, rXD);
(f) r is isomorphic to rx',D' for some X', D' G Ob (SS — Act).
Proof. Let us prove 1). (a) ^ ( b) By Lemma 1(c), (pa,x,d(f), ex, ев) G Homchu(ss-Act)(r, rXD). Since pa,x,d(f) is a monomorphism and ex, ев are isomorphisms in the category S S — Act, then (pa,x,d (f), ex, ев ) is a monomorphism in the category Chu(SS — Act).
(b) ^ ( b') Since (r, ex, eD) G HomChu(ss-Act)(r, rxD) is a monomorphism in the category Chu(SS — Act), then by Theorem 2, r is a monomorphism in the category S S — Act. Assuming w = r, we get (b').
(bf) ^ (c) Obviously.
(c) ^ (a) By Lemma 1(d), we have h9 ■ r = /. Since f is a monomorphism, then r is a monomorphism too. Thus, r is a separable Chu spaces.
Let us prove 2). Let r' G Homss-Act(A' xX', D'). Since r' is a separable Chu spaces, then by (b'), there exists w' G Homss-Act (A,D'X') such that (w', eX', eD') G HomChu(ss-Act)(r', rX'D'). Then
(w' ■ f,g, h) G Homchu(ss-Act)(r, rX'D').
By Lemma 1( d), we have w' ■ f = h9 ■ r. Since w' ■ f is a monomorphism, then is a monomorphism too. Thus, is a separable Chu spaces.
Let us prove 3). ( d) ^ ( e) By Lemma 1( c), we have
(r, ex, eD) G Homchu(ss-Act)(r, rXD).
Since r is a complete separable Chu spaces, it follows that r, ex, ев are isomorphisms. Hence ( r, ex, ев) is an isomorphism in the category Chu(SS — A ).
(e) ^ (f) Obviously.
(f) ^ (d) Let (f,g,h) £ Homchu(ss-Act)(r,rxD) is an isomorphism in the category Chu(SS — Act). Then f,g and h are isomorphisms in the category SS — Act. Therefore hg is an isomorphisms. By Lemma 1(d), we have f = h9 ■ r. Thus, r is an isomorphisms, i.e., r is a complete separable Chu spaces. □
6. Functors with values in the category Chu(SS — Act)
Consider the following functors: Pi : Chu(SS — Act) ^ (SS — Act), P2 : Chu(SS — Act) ^ (SS — Act)0, P3 : Chu(SS — Act) ^ (SS — Act), P23 : Chu(SS — Act) ^ (SS — Act)0 x (SS — Act) that map each Chu space r e Homss-Act(A x X, D) to the objects
Pi(A,X,D,r) = A, P2(A,X,D,r) = X, P3(A, X, D, r) = D, P23(A, X, D, r) = (X, D) and each morphism (f,g,h) e Homchu{ss-Act)(r,r') to the morphisms
Pi(f,9,h) = f, P2(f,g, h) = g, P3(f,g,h) = h, P23(f,g,h) = (g,h).
The fact that these are functors directly follows from the definition of composition of Chu morphisms.
Let Z be a category, F : Z ^ Chu(SS — Act) be a functor. By Fl, F2, F3, F23 = (F2, F3) we denote the functors acting as coordinates of F:
Fi = Pi о F : Z ^ (SS — Act), F2 = P2 о F : Z ^ (SS — Act)0,
F3 = P3oF : Z ^ (SS—Act), F23 = P23oF : Z ^ (SS—Act)0x(SS—Act).
Theorem 3. (on functors in Chu(SS — Act))
1) Let F : Z ^ Chu(SS — Act) be a functor. Then there are uniquely defined functors Fi : Z ^ SS — Act, F2 : Z ^ (SS — Act)0, F3 : Z ^ SS — Act such that for any object z e Ob(Z) and any morphism a e Homz(z, z'), we have
F (z) = (Fi(z),F2(z),F3(z),r(z)), where r(z) e Homss-Act(Fi(z) x F2(z),F3(z)), and
F(a) = (Fi(a),F2(a),F3(a)) e HomChu(ss-Act)(r(z),r(z')).
2) Let Fi : Z ^ (SS — Act), F2 : Z ^ (SS — Act)0, F3 : Z ^ (SS — Act) be the functors and for any z e Ob(Z), the morphism r(z) e Homss-Act(Fi(z) x F2(z),F3(z)) is fixed. Then the following conditions are equivalent:
(a) the mapping set by the equalities
F(z) = (Fi(z), F2(z),F3(z), r(z)), F(a) = (Fi(a), ^2(0), F3(a)) (1)
is a functor F : Z ^Chu(SS—Act), where z e O b(Z) anda e Homz (z ,z');
(b) for any a e Homz (z, z') we have
(Fi(a),F2(a),F3(a)) e HomChu(ss-Act)(r(z), r(z')). (2);
(c) for any a e Homz (z, z') we have
F3(a)F2(a) ■ i(z) = r(?) ■ Fi(o); (3);
(d) the family
W = (W(z) = K?) e Homss-Act(Fi(z),F3(z)F2(z)) | z e Ob(Z)}
is a homomorphism of functors W : Fi ^ F^2 = %ss o (F2, F3).
Proof. Let us prove 1). Since F(z) is a Chu space, it follows that F(z) = (A, X, D, r(z)) for some r(z) e HomSs-Act(A x X, D). So, by the notations above, A = Pi(F(z)) = Fi(z), X = P2(F(z)) = F2(z), D = Pi(F(z)) = F(z). If a e Homz(z, z') then
F(a) = (f,9,h) e HomChu(ss-Act)(r(z), r(z')).
Hence f = Pi(F(a)) = Fi(a), g = P2(F(a)) = F2(a), h = P(F(a)) = F3(a). Thus, F(a) = (Fi(a),F2(a),F3(a)). Let us prove 2). ( a) ^ (b) Obviously.
(b) ^ ( a) Since r(z) e Homss-Act(Fi(z) x F2(z), F(z)), it follows that F(z) e Ob(Chu(SS — Act)). By (2), we have
F(a) = (Fi(a),F2(a),F3(a)) e HomChu(ss-Act)(r(z), r(z')).
Let b e Homz(z',z''). Since Fi, F2 and F3 are functors, then by the definition of composition of Chu transform, we have
F(b oa) = (Fi(b oa),F2(b oa),F3(b o a)) =
= (Fi(b) ■ Fi(a),F2(a) ■ F2(b),F3(b) ■ F3(a)) = = ( Fi(b),F2(b),F3(b)) o (Fi(a),F2(a), F3(a)) = F(b) o F(a),
and F(ez) = (Fi(ez),F2(ez),F3(ez)) = (1Fi(z), 1f2(z), 1f3(z)) = 1f(z). Thus, F is a functor.
By Lemma 1( a), the conditions (b) and ( c) are equivalent. By definition of a morphism (natural transformation) of functors, the conditions ( c) and (d) are equivalent. □
7. Fundamental theorem on the functor H
Consider the following functors:
Hi = Uss : (SS — Act)0 x (SS — Act) ^ (SS — Act),
H2 : (SS — Act)0 x (SS — Act) ^ (SS — Act)0,
H3 : (SS — Act)0 x (SS — Act) ^ (SS — Act),
where H2(X,D) = X, H2(g,h) = g, H:i(X,D) = D, H3(g,h) = h for all X,D e Ob(SS — Act), g e Homss-Act(X', X), h e Homss-Act(D, D'). Then
rxD = Homss-Act(Hi(X, D) x Щ(Х, D), Щ(Х, D)),
and by Lemma 1(6), we have
(Hi(g,h),H2(g,h),H3(g,h)) = (h9,g,h) e Homchu(ss-Act)(rxD,rx'D).
Therefore the condition (b) of Theorem 3 is true for the category Z = (SS — Act)0 x (SS — Act) and functors Fi = Hi. Hence, by Theorem 3(a), the mapping given by the equalities
H (X,D) = (Dx ,X,D,rxD ),H (g,h) = (h9 ,g,h)
is a functor H :(SS — Act)0 x (SS — Act) ^ Chu(SS — Act).
Theorem 4. (on the functor H)
1) The functor H is full and faithful.
2) Let F : Z ^ Chu(SS — Act) be a functor given by the equality (1). There is a canonical functor homomorphism
V = [V(z) | z e Ob(Z)} : F ^ H о F23,
such that V(z) = {W(z),eF2(z),eF3(z)) e Homchu(ss-Act)(r(z),rF2(z)F3(z)),
where W(z) = ф) e Homss-Act(Fi(z),F3(z)F2(z)).
3) The functor H is right adjoint for the functor P23.
4) (a) Let z e Ob(Z). Then
V(z) is a monomorphism & W(z) is a monomorphism & F(z) is a separable Chu space;
V(z) is an isomorphism & W(z) is an isomorphism & F(z) is a complete separable Chu space.
(b) The functor homomorphism V : F ^ H о F23 is an isomorphism & F(z) is a complete separable Chu space for all z e Ob(Z).
Proof. 1) Let us show that H is a full and faithful functor, i.e., the mapping
Hom(ss-Act)°x(ss-Act)((X, D), (X', D')) ^ Homchu(ss-Act)(rxD, rx'D'),
such that H(g,h) = (ha,g,h), is bijective. The injectivity is obvious. We prove surjectivity. Let (f,g,h) £ Hom,chu(ss-Act)(rxD,rx'D'). Since (h9,g, h) £ Homchu(ss-Act)(rxD ,rx'D'), then by Lemma 1(d), we have f = h, i.e., H(g, h) = (f, g, h). Hence, the mapping H(g, h) ^ (h9,g, h) is bijective. Thus, the functor H is full and faithful.
2) There are equalities (H o F23)(z) = rF2(z)F((z);
(H o F23)(a) = (F3 (af2 (a), ^2(0)^3(0));
F(z) = r(z); F(a) = (^(a), ^(a), ^(a)).
To prove that V is a functor homomorphism, , it is necessary to prove the equality
V(z') o F(a) = (H o F23)(a) o V(z) (4)
for all a £ Homz(z, z').
Since V(z') o F(a) = (W(z') ■ F1(a),eF2(z') ■ F2(a),eFa(z') ■ F3(a)) and (H o F23)(a) o V(z) = (F3(a)F2(a) ■ W(z),F2(a) ■ eF2(z),F3(a) ■ eFs{z)), then the equality (4) means that the following three equalities are true:
W(z!) ■ Fx(a) = F3(a)F2(a) ■ W(z);
eF2(z') ■ F2(a) = F2(a) ■ eF2(z); &f3(Z') ■ ^3(a) = ^3(a) ■ &F({z).
The first of the equality coincides with equality (3) of Theorem 3 and therefore it is true, the second and the third equalities are obvious.
3) Let us use one of the standard properties of adjoint functors [4], and to do this, we will prove that the anjunction gives an unit and a counit, i.e. there are the functor homomorphisms : 1Chu{ss-Act) ^ H o P23, e : P23 o H ^ 1ss-Act°xss-Act, such that
He o -qH = 1H, SP23 o P23n = 1p2(, (5)
where the functor homomorphisms
He : H o P23 o H ^ H and -qH : H ^ H o P23 o H
are defined as follows:
(r}H)(X, D) = ri(H(X, D)) : H(X, D) ^ (H o P23)(H(X, D)),
(He)(X, D) = H(e(X, D)) : H((P23 o H)(X, D)) ^ H(X, D).
Note that P23 o H = 1(Chu(ss-Act))°xChu(ss-Act).
Define the counit e of adjunction as the identity homomorphism
e = [e(X,D) £ Hom{ss-Act)ox(ss-Act)((X,D), (X,D)) \
X,D £ Ob(SS — Act)},
e(X,D) = (ex, ). Obviously, the functor homomorphisms He and eP23 are the identity transformation of functors.
To define the units rj of an adjunction we apply the result of 2) to the case Z = Chu(SS — Act) and F = 1Chu{ss-Act) : Chu(SS — Act) ^ Chu(SS — Act). Therefore F23 = P23 and if r e Homss-Act(A x X,D), then H o P23,(r) = H(X, D) = rXD, V(r) = (r, ex, eD) e HomChu(ss-Act)(r, rxD) and V : 1chu(ss-Act) ^ H o P23 is a functor homomorphism. Suppose = V. Since fxD = eDx , it follows that
( r]H)(X,D) = V(rXD) = (eDx, ex, eD) =
= 1H(X,D) e HomChu{ss-Act)(H(X,D),H(X, D))
so that r]H : H ^ H = H o P23 o H is the identity transformation of functors. We also have
( P23v)(r) = P23(r, , eD) = (ex, eD) =
= 1P23(r) e Hom(sS-Act)°x(SS-Act)(P23(r),P23(r)).
Hence P23r] : P23 ^ P23 = P23 o H o P23 is the identity transformation of functors.
Thus, the functor homomorphisms He, rjH, eP23, P23r] are the identity transformation of functors, hence the equalities (5) are hold. Therefore the functor H is right adjoint for the functor P23.
4) directly follows from Theorem 3 and general properties of functor homomorphisms. □
8. Limits, products and coproducts in the category
Chu(SS — Act)
Theorem 5. (on limits) Let Z be a category, F : Z ^ Chu(SS — Act) be a functor such that F(z) is a complete separable Chu space for all z e Ob(Z). If every functor Z ^ SS — Act has a limit and every functor Z0 ^ SS — Act has a colimit, then there exists IimF that is complete separable Chu space.
Proof. By Theorem 3, there are functors Fi : Z ^ SS — Act, F2 : Z ^ (SS—Act)0, F3 :Z ^ SS—Act such that F(z) = (Fi(z), F2(z),F3(z), r(z)), F(a) = (Fi(a),F2(a),F3(a)), where
r(z) e HomSs-Act(Fi(z) x F2(z), F:i(z)), a e Homz(z,z').
By F21 : Z° ^ SS — Act we denote the functor given by the equalities F£(z) = F2(z) and F2(a) = F2(a) e Ham{SS-Act)o(F2(z),F2(z')) = Homss-Act(F2(z'),F2(z)), where a e Homz°(z', z). By the conditions of
Theorem, the functor F2 has a colimit, and the functor F3 has a limit, i.e., there are universal cones
^ = {p2(z) £ Hom(ss-Act)o(Fi(z),X)) \ * £ Ob(Z)},
P3 = {V3(z) £ Homss-Act(D,F3(z)) \ z £ Ob(Z)}
where the first cone is the colimit, the second cone is the limit. Thus, X = coUmF2, D = IimF3. By properties of dual categories,
P2 = {P2(z) £ Homss-Act(X,F2(z)) \ z £ Ob(Z)}
is the limit cone of the functor F2 such that X = IimF2. Hence, by the properties of the product of categories,
P23 = {(P2(Z),ifi3(Z)) £ Hom(sS—Act)0x(SS—Act)((X, D), F23(Z)) \ z £ Ob(Z)}
is the limit cone of the functor F23 = (F2, F3).
Since the functor H has a left adjoint, it translates the limit cone into the limit cone, i.e., H(p23) = {H(p23(z)) \ z £ Ob(Z)}, where H(p23(^)) = (p3(z)V2(z\p2(z),p3(z)) is the limit cone of the functor H o F23. In particular, H(X, D) = rxD = Iim(H o F23).
By Theorem 4, there is a canonical functor homomorphism V : F ^ H o F23. Since each F(z) is a complete separable Chu space, then V is an isomorphism of functors. Hence
{y( ^)-1 oH(p23(z)) £ Homchu(ss-Act)(rxD, F(z)) \ z £ Ob(Z)}
is the limit cone of the functor F. Therefore IimF = rxD. Since V(z) =
1 -1
(r(Z), 1 F2(z), 1F((z)) then V(Z) 1 = (r(Z) , 1F2(z), 1F((z)), i.e.,
V (z)-1 oH (p2 3 )(z) = (f(z)-1 ■P3(zr2 (z),p2 (z),P3(z)).
□
The proof of Theorem 6 implies the existence of the product in complete separable Chu spaces.
Theorem 6. Let n £ Homss-Act(Ai x Xi, Di) (i £ I) be the complete separable Chu spaces. The product of Chu spaces ri, i £ I, is the complete separable Chu space rxoDo with Chu transforms
((ri)-1 ■ P?, Qi, Pi) £ Homchu(ss-Act)(rx0D0, ri),
where X0 = U X^, D0 = ]\ Dj,, ^(s,Xj) = Xj,, pi(s,d) = d(i) for all
iei iei Xi £ Xi, d £ n Di, i £ F iei
Proof. Consider a discrete category Z such that objects are elements of the set L Then the family {n £ Homss-Act(Ai x Xi,Di) | i £ 1} is the same as functor F : Z ^ Chu(SS — Act) defined by equality F(i) = n, and the limit of the functor F is the product of the family. Therefore, the result being proved is a particular case of Theorem 5. □
The following theorem shows that in the category Chu(SS — Act) the coproducts exist for any Chu spaces.
Theorem 7. Let n £ Homss-Act(Ai x Xi,Di), i £ L The coproduct of the Chu spaces ri, i £ I, is the Chu space
r £ HomSS-Act(n Ai x П Xi, Ц Di)
iei iei iei with Chu transforms (fi, g^hi) £ HomChu(ss-Act)(П, r), where
r(s, (ai, x)) = n(s, (ai, x(i))), fi(s, ai) = ai, gi(s,x) = x(i), h (s ,di) = di
for all ai £ Ai, x £ П Xi, di £ Di, i £ L iei
Proof. Let i £ L The equalities
h(s, (n(s, ( ai, gi(s, x))))) = n(s, (ai, x(i))) =
= r(s, ( ai, x)) = r(s, (fi(s, ai), x))
for all ai £ Ai, x ^Xj, imply well-definability of the definition of the
iei
Chu transform (fi, g i,hi).
Let t £ HomSs-Act(B x Y, D), (f[, д[,^) £ Homchu(ss-Act)(n, t). By Theorem 4 [6], S-act Ц Ai with morphisms fi, i £ I, is a coproduct of iei
S-act Ai (i £ I), S-act Ц Di with morphisms hi, i £ I, is a coproduct
e
of S-acts Di (i £ I), and S-act П Xi with morphisms gi, i £ I, is a
e
product of S-acts Xi (i £ I) in the category SS — Act. Then there are unique morphisms f £ HomSs-Act(U. Ai,B), g £ HomSs-Act(Y,]\ Xi),
e e
h £ Homss-Aci(U Di, D) such that f[ = f ■ fi, g\ = ^ -g and h[ = h ■ h for e
all i £ I, i.e., f(s, ai) = fc(s,ai), g(s, y)(i) = g[(s, y) and h(s,d) = hhi(s,d) for all ai £ Ai, у £Y, d £ D, i £ L
Let us prove ( f ,g,h) £ HomChu(ss-Act)(r, t), that is, the equality is
hold h( s, r( s, (a, g(s ,y)))) = t(s, (f(s ,a), y)) for all a £ Ц Ai, у £Y. Since
e
(fi, 9%,hi) e Homchu(ss-Act) (n, r), then
r(s, (fi(s,a), g(s, y))) = h(s, n(s, (a, (^ ■ g)(s, y)))) for all a e Ai, y eY. Since hi = h ■ h and gi = gi ■ g, then
h( , ( , ( a, ( )))) =
= (h ■ hi)( s, n(s, ( a, (gi ■ g)(s, y)))) = hi( s, n(s, (a, gi(s, y))))
for all a e Ai, y e Y. Since (fi, gi,hi) e Homchu(ss-Act)(ri, t), then hi(s, n(s, (a, gi(s, y))) = t(s, (f[( s,a), y))) for all a e Ai, y e Y. Since f(s,a) = fi(s, a) then t(s, (fi(s,a), y)) = t(s, (f(s,a), y)) for all a e Ai, y eY. Thus,
h(s, r(s, ( a, g(s, y)))) = t(s, ( f ( s, a) x y))
for all a e U Ai, y eY. □
iei
9. Conclusion
In this paper, we study the category Chu(SS — Act). It is known [6] that the category SS — Act is Cartesian closed and the embedding functor S — Act ^ SS — Act has a left adjoint. Using this result, we prove the general properties of morphisms of Chu spaces and functors with a value in the category Chu(SS — Act) of Chu spaces over the category SS — Act. As a consequence, for the category Chu(SS — Act) the existence of coprod-ucts and some products is proved, monomorphisms and epimorphisms are characterized; in terms of this category the characteristics of separable and complete separable Chu spaces are given.
References
1. Barr M., Wells C. Category Theory for Computing Science. London, Prentice Hall Publ., 1995, 325 p.
2. Kilp M., Knauer U., Mikhalev A.V. Monoids, acts and categories. With applications to wreath products and graphs. Berlin, De Gruyter Publ., 2000. https://doi.org/10.1515/9783110812909
3. Kozhukhov I.B., Mikhalev A.V. Acts over semigroups, Fundamental and Applied Mathematics, 2020, vol. 23, no. 3, pp. 141-199. (in Russian)
4. Mac Lane S. Categories for the working mathematician. Graduate Texts in Mathematics, New York, Springer, 1998.
5. Skurikhin E.E. Presheaves of sets and actions of semigroups. Far Eastern Mathematical Journal, 2019, vol. 19, no. 1, pp. 63-74. (in Russian)
6. Skurikhin E.E., Stepanova A.A., Sukhonos A.G. Extensions of the category S-Act. Sib. Elektron. Mat. Izv., 2021, vol. 18, pp. 1332-1357. https://doi.org/10.33048/semi.2021.18.102
7. Stepanova A.A. S-acts over a Well-ordered Monoid with Modular Congruence Lattice. The Bulletin of Irkutsk State University. Series Mathematics, 2021, vol. 35, pp. 87-102. https://doi.org/10.26516/1997-7670.2021.35.87
8. Stepanova A.A., Chekanov S.G. Congruence-permutable S-acts. Siberian Mathematical Journal, 2022, vol. 63, no. 1, pp. 167-172. DOI: https://doi.org/10.1134/S0037446622010141
9. Stepanova A.A., Skurikhin E.E., Sukhonos A.G. Category of Chu spaces over S-Act category, Sib. Elektron. Mat. Izv., 2017, vol. 14, pp. 1220-1237. (in Russian) https://doi.org/10.17377/semi.2017.14.104
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11. Stepanova A.A., Skurikhin E.E,. Sukhonos A.G. Product of Chu spaces in the category of Chu(S-Act), Sib. Elektron. Mat. Izv., 2020 , vol. 17, pp. 1352-1358. (in Russian). https://doi.org/10.33048/semi.2020.17.099
Список источников
1. Barr M., Wells C. Category Theory for Computing Science. London : Prentice Hall, 1995. 325 p.
2. Kilp M., Knauer U., Mikhalev A. V. Monoids, acts and categories. With applications to wreath products and graphs. Berlin : De Gruyter, 2000.
3. Кожухов И. Б., Михалёв А. В. Полигоны над полугруппами // Фундаментальная и прикладная математика. 2020. Т. 23, № 3. C. 141-199.
4. Mac Lane S. Categories for the working mathematician, Graduate Texts in Mathematics. New York : Springer, 1998.
5. Скурихин Е. Е. Предпучки множеств и действия полугрупп // Дальневосточный математический журнал. 2019. Т. 19, № 1. C. 63-74.
6. Skurikhin E. E., Stepanova A. A., Sukhonos A. G. Extensions of the category S-Act // Sib. Elektron. Mat. Izv. 2021. Vol. 18. P. 1332-1357. https://doi.org/10.33048/semi.2021.18.102
7. Степанова А. А. S-полигоны над вполне упорядоченным моноидом с модулярной решеткой конгруэнций [S-acts over a Well-ordered Monoid with Modular Congruence Lattice] // Известия Иркутского государственного университета. Серия Математика. 2021. Т. 35. С. 87-102. https://doi.org/10.26516/1997-7670.2021.35.87
8. Степанова А.А., Чеканов С.Г. Конгруэнц-перестановочные полигоны // Сибирский математический журнал. 2022. Т. 63, № 1. С. 202-208. https://doi.org/10.33048/smzh.2022.63.114
9. Степанова А. А., Скурихин Е. Е., Сухонос А. Г. Категории пространств Чу над категорией полигонов // Сибирские электронные математические известия. 2017. Т. 14. С. 1220-1237. https://doi.org/10.17377/semi.2017.14.104
10. Степанова А. А., Скурихин Е. Е., Сухонос А.Г . Уравнители и коурав-нители в категориях пространств Чу над категорией полигонов // Сибирские электронные математические известия. 2019. Т. 16. C. 709-717. https://doi.org/10.33048/semi.2019.16.046
11. Степанова А. А., Скурихин Е. Е., Сухонос А. Г. Произведение пространств Чу в категории СЬи(8-Лс1) // Сибирские электронные математические известия. 2020. Т. 17. С. 1352-1358. https://doi.org/10.33048/semi.2020.17.099
Об авторах
Скурихин Евгений Евгеньевич,
д-р физ.-мат. наук, проф., Институт прикладной математики ДВО РАН, Владивосток, 690041, Российская Федерация; Дальневосточный федеральный университет, Владивосток, 690920, Российская Федерация, eeskur@gmail.ru, https://orcid.org/0000-0002-3649-5311
Степанова Алена Андреевна, д-р
физ.-мат. наук, проф., Дальневосточный федеральный университет, Владивосток, 690920, Российская Федерация, stepltd@mail.ru,
https://orcid.org/0000-0001-7484-4108
Сухонос Андрей Григорьевич,
канд. физ.-мат. наук, Дальневосточный федеральный университет, Владивосток, 690920, Российская Федерация, agsukh@mail.ru,
https://orcid.org/0000-0001-9253-5229
About the authors Evgeniy E. Skurikhin, Dr. Sci. (Phys.-Math.), Prof., Institute of Applied Mathematics FEB RAS, Vladivostok, 690041, Russian Federation, Far-Eastern Federal University, Vladivostok, 690920, Russian Federation, eeskur@gmail.ru, https://orcid.org/0000-0002-3649-5311
Alena A. Stepanova, Dr. Sci.
(Phys.-Math.), Prof., Far-Eastern Federal University, Vladivostok, 690920, Russian Federation, stepltd@mail.ru,
https://orcid.org/0000-0001-7484-4108
Andrey G. Sukhonos, Cand. Sci. (Phys.Math.), Far-Eastern Federal University, Vladivostok, 690920, Russian Federation, agsukh@mail.ru, https://orcid.org/0000-0001-9253-5229
Поступила в 'редакцию / Received 20.01.2023 Поступила после рецензирования / Revised 13.03.2023 Принята к публикации / Accepted 20.03.2023
