aa r X i v : . [ m a t h . C T ] S e p COMPACT OBJECTS IN CATEGORIES OF S -ACTS JOSEF DVOˇR ´AK AND JAN ˇZEMLIˇCKA
Abstract.
In this paper, the categorial property of compactness of an object, i. e. commutingof the corresponding Hom functor with coproducts, is studied in categories of S -acts and cor-responding structural properties of compact S -acts are shown. In order to establish a generalcontext and to unify approach to both of the most important categories of S -acts, the notion ofa category with unique decomposition of objects is defined and studied. Introduction
While great impact of theory of categories on ring and module theory is well known, theanalogical concept in the context of theory of monoids and acts of monoids on sets is significantlyless studied, however it seems to be promising and fruitful (as it is demonstrated in the monograph[11]).Recall that an object c of an abelian category closed under coproducts and products is said to be compact if the corresponding covariant functor Hom( c, − ) commutes with arbitrary direct sums i.e.there is a canonical isomorphism in the category of abelian groups Hom( c, ` D ) ∼ = ` Hom( c, D )for every system of objects D , where ` denotes a coproduct. In, particular a (right R -)module is small if it is compact in the category of all modules. As it is shown in [3] and in [13, 1 o ], smallmodules can be described in a natural way by the language of systems of submodules:The aim of the present paper is translating the notion of compactness in abelian categoriesto a more general context. The constitutive examples of such a generalization is provided byanalogy between (abelian) categories of modules over rings and (non-abelian) categories of actsover monoids. Lemma 1. [3, 13]
The following conditions are equivalent for a module M : (1) M is small, (2) if M = S i<ω M n for an increasing chain of submodules M n ⊆ M n +1 ⊆ M , then thereexists n such that M = M n , (3) if M = P i<ω M n for a system of submodules M n ⊆ M , n < ω , then there exists k suchthat M = P i Example 2. (1) A union of strictly increasing chain of the length κ , for an arbitrary cardinal κ ofuncountable cofinality, consisting of small (in particular finitely generated) submodules is small.(2) Every ω -generated uniserial module is small.A ring over which the class of all small right modules coincides with the class of all finitelygenerated ones is called right steady . Note that the class of all right steady rings is closed underfactorization [4, Lemma 1.9], finite products [16, Theorem 2.5], and Morita equivalence [7, Lemma1.7]. However ring theoretical characterization of steadiness is an open problem, it is known thatthe class of right steady rings contains various natural classes of rings including right noetherian[13, 7 ], right perfect [4, Corollary 1.6], right semiartinian of finite socle length [17, Theorem 1.5] Mathematics Subject Classification. Key words and phrases. compact object, monoid, S -act.This work is part of the project SVV-2020-260589. ountable commutative [13, 11 ], and abelian regular rings with countably generated ideals [18,Corollary 7].In section 2., as a useful common generalization of both of the most important categoriesof S -acts, the notion of a U D -category is introduced. Section 3 treats decompositions in U D -categories and then their necessary basic properties are shown, while in Section 4. a generalcomposition theory of projective objects in a U D -category is built. Finally, in Section 5., generalproperties of compact objects in a U D -category are shown and the application to categories of S -acts and the corresponding structural theorems follows. Furthermore, the more general propertyof autocompactness is studied, too.2. Axiomatic description of categories of acts Before we start to study common categorial properties of classes of acts over monoids, let usrecall needed terminology and notation.Let C be a category. Denote by Mor C ( θ, A ) the class of all morphisms A → B in C for everypair of objects A, B of C ; in case C is clear from the context, the subscript will be omitted. Amonomorphism (epimorphism) in C is a left (right)-cancellable morphism, i.e. a morphism µ suchthat µα = µβ ( αµ = βµ ) implies α = β . A morphism is a bimorphism, if it is both mono- andepimorphism. A category is balanced, if bimorphisms are isomorphisms (the reversed inclusionholds in general). An object θ is called initial provided | Mor( θ, A ) | = 1 for each object A . Thecategory is (co)product complete if the class of objects is closed under all (co)products.Let S = ( S, · , 1) be a monoid and A a nonempty set. If there is a mapping µ : S × A → A satisfying the following two conditions: µ (1 , a ) = a and µ ( s , µ ( s , a )) = µ ( s · s , a ) then A issaid to be a left S -act and it is denoted S A . For simplicity, µ ( s, a ) is often written as s · a or sa . A mapping f : S A → S B is a homomorphism of S -acts, or an S -homomorphism provided f ( sa ) = sf ( a ) holds for any s ∈ S, a ∈ A . In compliance with [11, Example I.6.5.] we denote by S − Act the category of all left S -acts with homomorphisms of S -acts and S − Act the category S − Act enriched by an initial object S ∅ . If the monoid S contains (necessarily unique) zeroelement 0, then the category of all left S -acts with homomorphisms of S -acts compatible with zeroas morphisms will be denoted by S − Act . Observe that { } is the initial object of the category S − Act .Recall that both of the categories S − Act and S − Act are complete and cocomplete [11,Remarks II.2.11, Remark II.2.22]. In particular, the coproduct of a system of objects ( A i , i ∈ I ) is(i) ` i ∈ I A i = ˙ S A i in S -Act by [11, Proposition II.1.8] and(ii) ` i ∈ I A i = { ( a i ) ∈ Q i ∈ I A i | ∃ j : a i = 0 ∀ i = j } in S -Act by [11, Remark II.1.16].Let C be a coproduct complete category with an initial object θ . An object A ∈ C is called indecomposable if it is not isomorphic to the initial object nor to any coproduct of two non-initialobjects. Note that cyclic acts present natural examples of indecomposable objects in both thecategories S − Act and S − Act . Nevertheless, the class of indecomposable acts can be muchlarger, e. g. the rational numbers form a non-cyclic indecomposable ( Z , · )-act.As we have declared, the main motivation of the present paper is to describe and explaincompactness properties of categories of acts over monoids in the general categorial language. Inparticular, we focus on the categories S − Act and S − Act . The key feature of both of thesecategories is the existence of unique decomposition of every object into indecomposable objectswhich is proved in [11, Theorem I.5.10] for the case of the category S − Act.First of all we list several natural categorial properties which ensure easy handling of thecategory, the uniqueness of a decomposition and provide existence condition, as well. Recall thata pair ( S, ν ) is said to be a subobject of an object A if S is an object and ν : S → A is amonomorphism.We say that C is a UD-category (uniquely decomposable) if the following conditions hold:(UD1) C is a coproduct complete balanced subcategory of the category of all sets, C contains aninitial object θ , for every object A the unique morphism θ → A is a monomorphism and | Mor( A, θ ) | ≤ UD2) for every morphism f ∈ Mor( A, B ) there exists the subobject ( f ( A ) , ι ) in C , where ι is theinclusion of f ( A ) into B (UD3) for every morphism f ∈ Mor( A, B ) and every subobject ( S, ν ) of B such that f ( A ) ⊆ ν ( S )there exists a morphism g ∈ Mor( A, S ) such that f = νg ,(UD4) for every system ( A i , ν i ) i ∈ I of subobjects of an object A both the sets T i ν i ( A i ) and S i ν i ( A i ) with respective inclusions form subobjects of A ,(UD5) if ( A, ( ν , ν )) is a coproduct of a pair of objects ( A , A ), then ν and ν are monomor-phisms and ν ( A ) ∩ ν ( A ) is isomorphic to θ ,(UD6) for every object A and every x ∈ A there exists an indecomposable subobject ( B, ν ) of A such that x ∈ B ⊆ A .Let us note that both categories of acts treated in this paper satisfy the previous axiomatics: Example 3. (1) Let S = ( S, · , 1) be a monoid. We show that all conditions (UD1)–(UD6) aresatisfied by S − Act, hence it is a UD-category.We have already mentioned that S − Act is a coproduct complete subcategory of the category ofsets. Furthermore, the empty act ∅ with the empty mapping represents an initial object and emptymap is a monomorphism since there is no morphism of a nonempty act to ∅ . Since monomorphismsare exactly injective morphisms, epimorphisms are surjective morphisms and isomorphisms arebijections, S − Act is (epi,mono)-structured hence reflective category (cf. [1, Section 14]), whichproves (UD1). The conditions (UD2), (UD3), (UD4) and (UD5) follow immediately from thedefinition of an act and (UD6) holds true since cyclic acts are indecomposable.(2) Let S = ( S , · , 1) be a monoid with a zero element 0. Then S − Act is also coproduct com-plete subcategory of the category of sets and the zero objects { } with the zero (mono)morphismforms an initial object of the category S − Act . Since there is exactly one (zero) morphism fromarbitrary object to the zero object, ( U D 1) holds true. A similar argumentation as in (1) showsthat S − Act satisfies also the conditions (UD2)–(UD6), i.e. it is a UD-category.3. Decomposition and coproduct We will suppose in the sequel that C denotes a UD-category. First, we prove a key observationthat the description of coproducts in both categories of acts [11, Proposition II.1.8, Remark II.1.16]can be easily generalized to any UD-category C .If M is a family of objects in a category A , the corresponding coproduct is designated ( ` M , ( ν M | M ∈ M )) and the product ( Q M , ( π M | M ∈ M )), recall that ν M and π M are said to be the structural morphisms of the coproduct and the product, respectively. Proposition 4. Let A = ( A i ) i ∈ I be a set of objects in C and ( ` A i , ( ν i ) i ) be the coproduct of A with corresponding colimit structural morphisms. Then ` A i = S i ν i ( A i ) .Proof. Let us denote by ι the inclusion monomorphism ensured by (UD2). Since ( S i ν i ( A i ) , ι )is a subobject by (UD4), there exists a morphism µ j ∈ Mor( A j , S i ν i ( A i )) by (UD3) such that ιµ j = ν j for each j . Using the universal property of the colimit, we obtain a morphism ϕ suchthat ϕν j = µ j for every j , i.e. the left square of the diagram A i ν i −−−−→ ` A i ` A i (cid:13)(cid:13)(cid:13) y ϕ x ι A i µ i −−−−→ S ν i ( A ) S ν i ( A )commutes in C and we show that the right square commutes as well.Since ιϕν j = ιµ j = ν j for every j , we get again by the colimit universal property that ιϕ = id ` A i , hence the inclusion ι is onto ` A i , which finishes the proof. (cid:3) Let A be an object, A j ⊂ A a subobject and ι j the inclusion of A j into A for each j ∈ J . Wesay that ( A j , j ∈ J ) is a decomposition of A if ( A j , ι j ) is a subobject for each j ∈ J and ( A, ι j ) j ∈ J is the coproduct of the family ( A j , j ∈ J ).The following assertion describes a natural decomposition of a coproduct in C . emma 5. Let A = ( A j , j ∈ J ) be a system of objects in C and ( ` A j , ( ν j ) j ) the coproduct of A .Then ( ν j ( A j ) , j ∈ J ) is a decomposition of ` A j .Proof. Put A = ` A j and for every j denote by ι j the inclusions ν j ( A j ) ֒ → A and by ˜ ν j ∈ Mor( A j , ν j ( A j )) the morphism satisfying ν j = ι j ˜ ν j which exists by (UD3). Since maps ˜ ν j arefor all j surjective, they are epimorphism in C by (UD1). We need to prove that ( A, ( ι j ) j ) is acoproduct of ( ν ( A j ) , j ∈ J ).Suppose that B is an arbitrary object and ρ j ∈ Mor( ν j ( A j ) , B ) is a morphism for each j ∈ J .Fix ϕ ∈ Mor( A, B ) satisfying the property ρ j ˜ ν j = ϕν j for each j ∈ J , which exists by the universalproperty of the coproduct ( A, ( ν j ) j ). It remains to show that ϕ is the unique morphism such that ρ j = ϕι j for all j .Since ρ j ˜ ν j = ϕν j = ϕι j ˜ ν j and ˜ ν j is an epimorphism, we get the desired equality ρ j = ϕι j foreach j . Finally, if we have any morphism ˜ ϕ such that ρ j = ˜ ϕι j for each j , then ρ j ˜ ν j = ˜ ϕι j ˜ ν j = ˜ ϕν j for each j , hence ˜ ϕ = ϕ by the universal property of the coproduct ( A, ( ν j ) j ). (cid:3) It is well-known that θ ` A ∼ = A for every object A and there is a canonical isomorphism ` i ∈ I ( ` A i ) ∼ = ` (cid:0)S i ∈ I A i (cid:1) for every system of sets of objects A i , i ∈ I in any coproduct completecategory with an initial object θ .As a straightforward consequence of the previous lemma we obtain an important property ofdecompositions in UD-category. Lemma 6. Let A ∈ C be an object and ( A i , i ∈ I ) a system of disjoint sets of subobjects of A .The following conditions are equivalent: (1) for each i ∈ I , the set A i is a decomposition of an object B i and the family ( B i , i ∈ I ) isa decomposition of A (2) S i ∈ I A i is a decomposition of A . The following elementary observation will appear useful for further dealing with decompositionsof objects in a general UD-category. Lemma 7. An initial object θ of a balanced category has no proper subobjects, i.e. ν is an iso-morphism for any subobject ( S, ν ) of θ .Proof. Let ϑ S denote the unique morphisms of θ into S . Then id θ = νϑ S by uniqueness of theendomorphism of θ thus ν is an epimorphism. Since ν is a monomorphism by the definition, it isan isomorphism (cid:3) Since the morphism of an initial object to an arbitrary object is a monomorphism by (UD1) andsince a UD-category is balanced, every object A of C contains as a subobject with the inclusion auniquely defined isomorphic image of the initial object, denote it by θ A . Lemma 8. Let A ∈ C , S ⊆ A and ι the inclusion of S into A . If ( S, ι ) is a subobject of A , then θ S = θ A and θ A ( S whenever θ A = S .Proof. Denote by ϑ O the unique morphisms of θ into an object O . Since ιϑ S = ϑ A , we get that θ S = θ A and so θ A ⊆ S . (cid:3) Lemma 9. Let A ∈ C and A j ⊂ A be a subobject with the inclusion into A for every j ∈ J . Then ( A j , j ∈ J ) is a decomposition of A if and only if A = S j A j and A i ∩ S j = i A j = θ A for each i ∈ J .Proof. Let ( A j , j ∈ J ) be a decomposition of A . Then A = S j A j by Proposition 4. Since { A i , S j = i A j } is a decomposition of A by Lemma 6, (UD3) and (UD4), we get that A i ∩ S j = i A j ∼ = θ by (UD5). Thus A i ∩ S j = i A j = θ A by Lemma 8.In order to prove the reverse implication let us suppose that A = S j A j , and A i ∩ S j = i A j = θ A for each i ∈ J and ( ` j A j , ( ν j ) j ) is a coproduct of ( A j , j ∈ J ). Then there exists a morphism ϕ ∈ Mor( ` j A j , A ) such that ϕν j are inclusions by the universal property of the coproduct. Hence is onto because A = S j A j . Let ϕ ( a ) = ϕ ( b ) for elements a and b . Then there are indeces j , j ∈ J and elements ˜ a ∈ A j , ˜ b ∈ A j for which a = ν j (˜ a ), b = ν j (˜ b ) by Proposition 4, hence˜ a = ϕ ( ν j (˜ a )) = ϕ ( ν j (˜ b )) = ˜ b, which proves that ϕ is an injective map. Since ϕ is bijective morphism in a balanced subcategoryof the category of sets, hence monomorphism and epimorphism at the same time, it is an isomor-phism. As ( A, ( ϕν j ) j ) is a coproduct of ( A j , j ∈ J ) and all ϕν j are inclusions, ( A j , j ∈ J ) is adecomposition of A . (cid:3) Note that the argument of the reverse implication depends strongly on the fact that C is asubcategory of the category of sets by (UD1).We say that an object A in a category with an initial object is indecomposable if for every pairof objects A , A such that A is isomorphic to A ` A it holds that A or A is isomorphic toan initial object.Note that the definition of an indecomposable object in a UD-category reflects the definitionof such an object in categories of acts. The natural description of an indecomposable object isformulated in the next assertion: Proposition 10. Let A be an object in C . Then the following conditions are equivalent: (1) A is indecomposable, (2) for every decomposition ( A , A ) of A either A = θ A or A = θ A , (3) for every decomposition ( A j , j ∈ J ) of A there exists i ∈ J such that A j = θ A j for each j = i .Proof. (1) ⇒ (2) If ( A , A ) is a decomposition of A , then A ∼ = A ` A , hence either A ∼ = θ andso A = θ A = θ A or A ∼ = θ and so A = θ A = θ A by Lemma 8.(2) ⇒ (1) Let A ∼ = A ` A and ν , ν be structural morphisms of the coproduct. Then thepair ( ν ( A ) , ν ( A )) forms a decomposition of A by Lemma 5, hence either ν ( A ) or ν ( A ) isisomorphic to θ . Suppose w.l.o.g. that ν ( A ) ∼ = θ and denote by ι the inclusion of ν ( A ) into A , which is a morphism by (UD2), and denote by µ ∈ Mor( A , ν ( A )) a morphism satisfying ν = ι µ which exists by (UD3). Since ν is monomorphism by (UD5), µ is monomorphism aswell. Furthermore µ is epimorphism as ν ( A ) is isomorphic to the initial object. Hence µ isisomorphism, since C is a balanced category, which implies that A ∼ = ν ( A ) ∼ = θ .(2) ⇔ (3) The direct implication follows from Lemma 6 and the fact that any coproduct of initialobjects is isomorphic to initial object. The reverse implication is clear. (cid:3) Lemma 11. Let ( A i , i ∈ I ) be a family of subsets of an object A of the category C such that forevery i ∈ I , A i is an indecomposable object and together with the inclusion into A forms a subobjectof A . If T i ∈ I A i = θ , then A ′ = S i ∈ I A i with inclusion forms an indecomposable subobject of A .Proof. We repeat the argument of the proof of [11, Lemma I.5.9]. Since A ′ is a subobject of A by (UD4) and θ A = θ A ′ , we may suppose w.l.o.g. that A = A ′ = S i ∈ I A i . Assume that ( B , B )is a decomposition of A such that B i = θ A for both i = 0 , 1. Since θ A ⊆ C for each subobject C with inclusion of the object A and ( B ∪ B ) ∩ T i ∈ I A i = T i ∈ I A i = θ A , there exists j suchthat θ A ( B j ∩ T i ∈ I A i ⊆ A i , say j = 0. Furthermore, as S i ∈ I A i = A , there exists i ∈ I with B ∩ A i = θ A , hence θ A ( B ∩ A i . Hence ( B ∩ A i , B ∩ A i ) forms a nontrivial decomposition ofan indecomposable object A i by Lemma 9, a contradiction. (cid:3) Now we can formulate a version of [11, Theorem I.5.10] valid in a general UD-category: Theorem 12. Every nonititial object A in C a has a unique decomposition into indecomposableobjects.Proof. For a ∈ A \ θ A consider the set I a = { U | ( U, ⊆ ) is an indecomposable subobject of A and a ∈ U } and put A a = S I a . Then A a is indecomposable object by Lemma 11. urthermore, if a = b then either A a = A b , or A a ∩ A b = θ A . Indeed, let A a ∩ A b = θ A , take z ∈ A a ∩ A b \ θ A , which exists by Lemma 7 and (UD4) and consider the indecomposable object A z . Since z ∈ A a , we have A a ∈ I z , hence A a is a subobject of A z , similarly for b and viceversa,therefore A z = A a = A b .Note that for every a ∈ A there exists indecomposable subobject of A which contains a by(UD6), hence a ∈ A a . Moreover, as A is not isomorphic θ , we get that A = S a ∈ A \ θ A A a , and wehave proved that the representative set of subobjects of the form A x is the desired decomposition.It remains to show uniqueness of the decomposition. Let ( A i | i ∈ I ) and ( B j | i ∈ J ) be twodecompositions into indecomposable objects. By applying Lemma 9 and (UD4) we get ( A i ∩ B j | j )is a decomposition of the indecomposable object A i for each i ∈ I , hence there exists j suchthat A i = B j . A symmetric argument proves that there exists a bijection β : I → J for which A i = B β ( i ) which finishes the proof. (cid:3) Projective objects Recall that C is a UD-category. We say that an object P ∈ C is projective , if for any pair ofobjects A, B ∈ C and a pair of morphisms π : A → B , α : P → B , where π is an epimorphism,there exists a morphism α : P → A in C such that α = πα , i.e. any diagram P y α A π −−−−→ B in C with π an epimorphism, can be completed into a commutative diagram P P α y y α A π −−−−→ B. Lemma 13. The coproduct of a family ( P i , i ∈ I ) of projective objects of C is projective.Proof. Let the projective situation ` P i y α A π −−−−→ B be given.For each i ∈ I consider the structural monomorphism ν i : P i → ` P i , which gives P i P i y ϕ i y αν i A π −−−−→ B, where ϕ i : P i → A is obtained from the projectivity of P i ; the family ( ϕ i , i ∈ I ) induces the uniquemorphism ϕ : ` P i → A with ϕν i = ϕ i . By Proposition 4, each x ∈ ` P i can be written as ν i ( y )for some (not necessarily unique) i ∈ I and y ∈ P i , hence πϕ ( x ) = πϕν i ( y ) = αν i ( y ) = α ( x ). (cid:3) Lemma 14. In C , if P = ` I P i is projective then each P i is projective.Proof. By induction it is enough to prove for any pair of objects P , P that P is projective if P ` P is projective.Let the projective situation P y α A π −−−−→ B e given.Denote by ˜ α : P ` P → B ` P the coproduct of morphisms α : P → B and 1 P : P → P ,similarly denote by ˜ π : A ` P → B ` P the coproduct of morphisms π : A → B and 1 P : P → P , which are uniquely determined by the universal properties of the coproducts P ` P and A ` P . It is easy to compute applying the universal property of of the coproduct B ` P that ˜ π is epimorphism since both π and 1 P are epimorphisms.Hence we obtain another projective situation: P ` P y ˜ α A ` P π −−−−→ B ` P By the assumption there exists a morphism ϕ such that ˜ πϕ = ˜ α .Let λ X : X → P ` P , for X ∈ { P , P } , µ X : X → A ` P (for X ∈ { A, P } ) and ν X : X → B ` P (for X ∈ { B, P } ) be structural coproduct morphisms (all mono by UD5).Let us show that ϕλ P ( P ) ⊆ µ A ( A ). Denote by S := ϕλ P ( P ) ∩ µ P ( P ) and by ι the inclusion S → A ` P . Note that ( S, ι ) is a subobject of A ` P by (UD2) and (UD4), Then by (UD3) thereexist monomorphisms σ i : S → P i , i = 0 , P λ P −−−−→ P ` P P ` P x σ y ϕ y ˜ α S ι −−−−→ A ` P π −−−−→ B ` P y σ x µ P x ν P P P P . Hence we get: ˜ πι ( S ) = ˜ αλ P σ ( S ) = ν B ασ ( S ) ⊆ ν B ( B ), where the last equality holds bythe definition of coproduct morphism ˜ α , i.e. because ˜ αλ P = ν B α . On the other hand, we alsoget ˜ πι ( S ) = ν P σ ( S ) ⊆ ν ( P ). By Lemmas 5 and 9 we obtain ˜ πι ( S ) ⊆ θ B ` P , which byLemma 7 turns into ˜ πι ( S ) = θ B ` P . In conclusion, since ˜ πι ( S ) = ν P σ ( S ) and ν P , σ are bothmonomorphisms, S itself is a subobject of the zero object, i.e. S ∼ = θ . As θ µ P ( P ) is a unique zerosubobject of µ P ( P ), θ µ P ( P ) = θ A ` P , henceforth the desired ϕλ P ( P ) ⊆ µ A ( A ).In consequence, by (UD3) there exists a morphism τ : P → A such that µ A τ = ϕλ P ; therefore˜ πϕλ P = ˜ πµ A τ = ν B πτ and on the other hand ˜ πϕλ P = ˜ αλ P = ν B α . Since by (UD5) themorphism ν B is mono, we have α = πτ . (cid:3) Now we are ready to prove an important property of UD-categories: Theorem 15. An object of a UD-category is projective if and only if it is isomorphic to a coproductof indecomposable projective objects.Proof. If an object is projective, it possesses a decomposition by Theorem 12, which consists ofprojective objects by Lemma 14. The reverse implication follows immediately from Lemma 13. (cid:3) Let A and B be objects of C . Recall that B is a retract of A if there are morphisms f ∈ Mor( A, B )and g ∈ Mor( B, A ) such that f g = 1 B . The morphism f is called a retraction and f a coretraction .Note that each retraction is an epimorphism and each coretraction is a monomorphism. An object G of a category is said to be generator if for any object A ∈ C there exists an index set I and anepimorphism π : ` i ∈ I G i → A where G i ≃ G . Lemma 16. If C contains a generator G , then every indecomposable projective object is a retractof G . roof. We generalize arguments of [11, Propositions III.17.4 and III.17.7].Let P be an indecomposable projective object. Since G is a generator, there are coproduct ` i G i of objects G i ∼ = G with structural morphisms ν i ∈ Mor( G i , ` i G i ) and an epimorphism π ∈ Mor( ` i G i , P ). Moreover, there exists a (mono)morphism γ ∈ Mor( P, ` i G i ) for which πγ = 1 P due to projectivity of P . As ( P ∩ ν i ( G i ) , i ) forms a decomposition of indecomposable P ∼ = γ ( P )by Lemmas 9 and 7, there is i such that γ ( P ) ⊆ ν i ( G i ). Then by (UD3) there exists a morphism ϕ ∈ Mor( P, G i ) such that ν i ϕ = γ . Thus πν i ϕ = πγ = 1 P which shows that πν i is the desiredretraction. (cid:3) Compact objects Now we are ready to translate the concept of compactness to the context of a UD-category C .Let C be an object of the UD-category C , A = ( A i , i ∈ I ) be a family of objects of C and denoteby ν i : A i → ` I A i the corresponding coproduct structural morphisms. Let us define a naturalmorphism in the category Set of all setsΨ C A : a i Mor R ( C, A i ) → Mor R ( C, a i A i )to be the unique morphism such that the following square, with µ i : Mor ( C, A i ) → ` I Mor ( C, A i )being the coproduct structural inclusion, is commutative for all i ∈ I :Mor ( C, A i ) µ i −−−−→ ` I Mor ( C, A i )Mor ( ν i ,A i ) y Ψ C A y Mor ( C, ` I A i ) Mor ( C, ` I A i )Since Mor presents a forgetful functor of the category C into the category Set and coprod-ucts of objects in Set are isomorphic to disjoint unions of the corresponding objects, we have ` I Mor ( C, A i ) = ˙ S Mor ( C, A i ) and we can describe Ψ C A explicitly as Ψ C A ( α ) = ν i α for the index i satisfying α ∈ Mor( C, A i ). Lemma 17. Let C be an object, I be an index set of at least two elements, A = { A i | i ∈ I } afamily of objects in the category C , and α, β ∈ ` I Mor ( C, A i ) . (1) If α = β and Ψ C A ( α ) = Ψ C A ( β ) , then there exist i = j such that α ∈ Mor ( C, A i ) , β ∈ Mor ( C, A j ) and ν i α ( C ) = ν j β ( C ) = θ ` A i . (2) If i, j ∈ I such that α ( C ) = θ A i and β ( C ) = θ A j for some α ∈ Mor ( C, A i ) and β ∈ Mor ( C, A j ) , then Ψ C A ( α ) = Ψ C A ( β ) . (3) Ψ C A is injective (i.e. it is a monomorphism in the category Set) if and only if Mor ( C, θ ) = ∅ .Proof. Let us denote A = ` I A i .(1) If there exists i for which α, β ∈ Mor ( C, A i ), then ν i α = ν i β , hence α = β by (UD5).It means the hypothesis α = β that implies there exists i = j such that α ∈ Mor ( C, A i ), β ∈ Mor ( C, A j ) and ν i α ( C ) = Ψ C A ( α )( C ) = Ψ C A ( β )( C ) = ν j β ( C ) ⊆ ν i ( A i ) ∩ ν j ( A j ) . Since ν i ( A i ) ∩ ν j ( A j ) = θ A by Lemmas 5 and 9, we get ν i α ( C ) = ν j β ( C ) = θ A by Lemma 7.(2) Since θ ∼ = θ A i ∼ = θ A j ∼ = θ A is the initial object of the category C , we obtain that ν i ( θ A i ) = θ A = ν j ( θ A j ), hence Ψ C A ( α )( C ) = ν i α ( C ) = ν j β ( C ) = Ψ C A ( β )( C ) = θ A again by Lemmas 5, 7 and 9. It implies that both morphisms Ψ C A ( α ) , Ψ C A ( β ) can be viewed aselements of Mor( C, θ ). As | Mor( C, θ ) | ≤ C A ( α ) = Ψ C A ( β ).(3) If Mor ( C, θ ) = ∅ , there exists α i ∈ Mor( C, A i ) such that α i ( C ) = θ A i for all i ∈ I . ThusΨ C A ( α i ) = Ψ C A ( α j ) for all i, j ∈ I by (2) which implies that Ψ C A is not injective.On the other hand, if Ψ C A is not injective, then there exists i and α ∈ Mor ( C, A i ) such that ν i α ( C ) = θ A by (1). As θ A ∼ = θ , there exists a morphism in Mor ( C, θ ) by (UD3). (cid:3) ince there is no morphism of a nonempty act C into the empty act ∅ , all mappings Ψ C A areinjective in the category S − Act by Lemma 17(3), similarly to the case of abelian categories (cf[9, Lemma 1.3]). Applying the same assertion, we can see that it is not the case of the category S − Act . Example 18. If S = ( S , · , 1) is a monoid with a zero element (for example ( Z , · , C is anact and A = { A i | i ∈ I } is a family of acts contained in the category S − Act such that |A| ≥ C A is not injective by Lemma 17(3).In particular, if we put C = A i = { } for every i ∈ I , then | ` I Mor ( C, A i ) | = | I | and | Mor ( C, ` I A i ) | = 1, so the mapping Ψ C A glues all morphisms of the arbitrary large set ` I Mor ( C, A i ).Using the notation of the mapping Ψ C A we are ready to define compact objects in UD-categories.Let D be a class of objects of the category C and denote by D ` = { ` i D i | D i ∈ D} the classof all coproducts of all families of objects of D .We say that an object C is D -compact (or compact with respect to D ), if the morphism Ψ C A issurjective for every subsystem A of objects of the class D and C is compact if it is O C -compactfor the class O C of all objects of the category C . Finally, an object C is called autocompact , if itis { C } ` -compact. Observe that every compact object is D -compact for an arbitrary class D , inparticular, it is autocompact. Theorem 19. The following conditions are equivalent for an object C ∈ C and a class of objects D : (1) C is D -compact, (2) for every pair of objects A ∈ D and , A ∈ D ` and every morphism ϕ : C → A ` A there exists i ∈ { , } such that ϕ ( C ) ⊆ ν i ( A i ) where ν i is the coproduct structuralmonomorphism, (3) any morphism f : C → ` i ∈ I A i into a coproduct of objects A i ∈ D factorizes through oneof the structural monomorphisms ν i : A i → ` i ∈ I A i (i.e. there exists i such that such that f ( C ) ⊆ ν i ( A i ) ).Proof. (1) ⇒ (3) Let A = { A i | i ∈ I } be a family of objects of the class D and f ∈ Mor (cid:0) C, ` i ∈ I A i (cid:1) .Since C is D -compact, the mapping Ψ C A is surjective by the definition, hence there exists i and α ∈ Mor ( C, A i ) such that f = ν i α .(3) ⇒ (2) It is obvious.(2) ⇒ (1) Suppose again that A = { A i | i ∈ I } is a family of objects of D , fix A = ` i ∈ I A i with ν i structural morphisms, and f ∈ Mor ( C, A ). By Lemma 5 ( ν i ( A i ) , i ∈ I ) forms a decompositionof the coproduct A , hence ν i ( A i ) ∩ S j ∈ I \{ i } ν j ( A j ) = θ A for each i by Lemmas 7 and 9. Notethat B = ` j ∈ I \{ i } A j ∈ D ` and S j ∈ I \{ i } A j is a morphic image of the structural morphism ofthe coproduct A ∼ = B , hence either f ( C ) ⊆ ν i ( A i ) or f ( C ) ⊆ S j ∈ I \{ i } ν j ( A j ) for each i by thehypothesis and Proposition 4.If there exists i for which f ( C ) ⊆ ν i ( A i ), then f is an element of the image of Ψ C A by (UD3)and we are done.Assuming f ( C ) ⊆ S j ∈ I \{ i } ν j ( A j ) for every i we get that ν i ( A i ) ∩ f ( C ) ⊆ ν i ( A i ) ∩ [ j ∈ I \{ i } ν j ( A j ) ⊆ θ A for each i by Lemma 9, hence f ( C ) = θ A by Lemma 7. Since θ A ∼ = θ A i is the initial object, thereexists g ∈ Mor( C, A i ) such that g ( c ) = θ A i and Ψ C A ( g ) = f for arbitrary i ∈ I , hence Ψ C A issurjective. (cid:3) Let us reformulate the Theorem 19 for the particular (but important) case of compactness: Corollary 20. The following conditions are equivalent for an object C of C : (1) C is compact, (2) for every pair of objects A and A and every morphism ϕ : C → A ` A there exists i ∈ { , } such that ϕ ( C ) ⊆ ν i ( A i ) where ν i is the coproduct structural monomorphism, any morphism f : C → ` i ∈ I A i for any family of objects A = { A i | i ∈ I } into a coproductfactorizes through one of the structural monomorphisms ν i : A i → ` i ∈ I A i . In order to obtain useful characterization of compact objects in a general UD category we saythat an object B is said to be an morphic image of an object A if there is a surjective morphism A → B (recall that UD-category is a subcategory of the category Set, hence a concrete category).Namely, we observe that compact objects in the category C are precisely objects whose everymorphic image is indecomposable: Proposition 21. An object C of the category C is compact if and only if every morphic image of C is indecomposable,Proof. ( ⇒ ) Let π : C → C be a surjective morphism and ( A , A ) a nontrivial decomposition of C . Then π is not an element of the image of Ψ C { A ,A } by Lemma 9, hence C is not compact.( ⇐ ) If C is not compact, then there exists a pair of objects A and A and a morphism ϕ ∈ Mor ( C, A ` A ) such that ϕ ( C ) * ν i ( A i ) by Corollary 20. Hence ( ϕ ( C ) ∩ A , ϕ ( C ) ∩ A ) isa nontrivial decomposition of ϕ ( C ) by Lemmas 5 and 9. (cid:3) As an easy consequence of the last claim is that every compact object in C is indecomposable.In a similar fashion the compactness (smallness) property originally studied in the area of(left R -)modules has been defined, the notion of self-smallness as a generalization of the propertyof being finitely generated can be transferred as the notion of an autocompact object to U D -categories and specially to those of S -acts. (see e.g. [2], [6]). The following useful characterizationof autocompactness presents another consequence of Theorem 19: Corollary 22. The following conditions are equivalent for an object C ∈ C : (1) C is autocompact, (2) for every morphism f ∈ Mor (cid:0) C, ` i ∈ I C i (cid:1) where C i ∼ = C for each i ∈ I there exists i suchthat such that f ( C ) ⊆ ν i ( C i ) where ν i is the coproduct structural monomorphism. Using the same arguments as in direct implication of Proposition 21 we get a necessary conditionof autocompact objects: Lemma 23. Autocompact objects in the category C are indecomposable.Proof. Assume the autocompact object C decomposes into C = C ∪ C . Consider the identitymorphism ι : C → C ∪ C ∼ = C ` C . Then either C = ι ( C ) ⊆ C or C = ι ( C ) ⊆ C byCorollary 22 and Lemma 9, hence C is indecomposable. (cid:3) Proposition 24. For an autocompact object C ∈ C and an endomorphism f ∈ Mor ( C, C ) , theobject D = f ( C ) is autocompact, too.Proof. Suppose D is not autocompact. Then by Corollary 22 there is a morphism g : D → ` i ∈ I D i such that D i ∼ = D and g ( D ) * ν i ( D i ), hence g ( D ) ∩ D i = θ D i for each i . Since g ( D ) ∩ D i is asubobject of C i ∼ = C with the corresponding inclusion morphism ν i , there exists a coproduct ofmorphisms ν = ` i ∈ I ν i ∈ Mor (cid:0)` i ∈ I D i , ` i ∈ I C i (cid:1) such that the composition C f ։ D g → a i ∈ I D i ν → a i ∈ I C i contradicts the autocompactness of C . (cid:3) Categories of S -acts Let S = ( S, · , 1) be a monoid (or a monoid with zero) through the whole section. Recall thatfor S both categories S − Act and S − Act of S -acts are UD-categories by Example 3. We will usebasic properties of these categories of acts summarized in axiomatics (UD1)–(UD6) freely in thesequel. For standard terminology concerning the theory of acts we refer to the monograph [11]. .1. Compact acts. The following consequence of Corollary 20 shows that reverse implicationof [11, Lemma I.5.36] holds true. Lemma 25. Compact objects in the category S − Act are precisely indecomposable objects.Proof. Since every epimorphic image of an indecomposable act in the category S − Act is againindecomposable by [11, Lemma I.5.36], we obtain the claim immediately from Corollary 20. (cid:3) Recall that a left S -act A is called cyclic if there exists a ∈ A for which Sa = { sa | s ∈ S } = A ,and A is called locally cyclic if for any pair a, b ∈ A there exists c ∈ A such that a, b ∈ Sc .Since cyclic acts are locally cyclic and that locally cyclic acts are indecomposable, we obtain animmediate consequence of Lemma 25: Corollary 26. Every locally cyclic left act is compact in the category S − Act . Furthermore, we prove a sufficient condition of compactness for both considered categories acts. Proposition 27. Every cyclic left act is compact in both categories S − Act and S − Act .Proof. We only need to prove the second claim, since the class of locally cyclic S -acts containsthe class of cyclic ones. Now, since a factor of a cyclic act is cyclic, hence indecomposable, theProposition 21 gives us the result in the category S − Act . (cid:3) The corresponding variant of Lemma 25 as the criterion of compactness in the category S − Act will need to deal with all factors of an act, namely, a compact object in category S − Act areprecisely objects whose every morphic image is indecomposable by Proposition 21.The following example shows that in case of the category S − Act the implication in Proposi-tion 27 cannot be inverted in general: Example 28. Let Z = ( Z , · , 1) be a monoid with zero.(1) Consider the (left) Z -act A = 2 Z ∪ Z . Then A is an indecomposable act which is notcompact in the category S − Act . Indeed, if we consider the morphism f : A → Z given by f ( a ) = a mod 6, then the image f ( A ) = { , , } ∪ { , } decomposes, hence it is not compactby Proposition 21.(2) Every abelian group is compact in the category Z − Act since every Z -subact contains 0.More generally, for a monoid S with zero, any A ∈ S − Act is also an object of S − Act, it becomesindecomposable, hence compact by Lemma 25.In compliance with [11, Definition 4.20] recall that for a subact B of an act A the Rees congruence ρ B on A is defined by setting a ρa if a = a or a , a ∈ B . The corresponding factor act A/B is called Rees factor of A by B then. Lemma 29. Let A be an act of the category S − Act with two subacts A and A . If A = A ∪ A and A i \ ( A ∩ A ) = ∅ for both i = 1 , , then A is not compact in S − Act .Proof. Consider the projection π of A onto the Rees factor A/ ( A ∩ A ), which is decomposableinto π ( A ) ` π ( A ). Now use Corollary 20. (cid:3) A subact B of a left S -act A (in S − Act or S − Act ) is called superfluous if B ∪ C = A for anyproper subact of A (see [12, Definition 2.1]). An act is called hollow if each of its proper subactsis superfluous (see [12, Definition 3.1]). Note that the situation of Lemma 29 is precisely that ofnon-hollow acts. Proposition 30. An S -act A is compact in the category S − Act if and only if it is hollow.Proof. Suppose A is hollow and it is not compact, i.e. there is a decomposable factor π ( A ) = A ` A by Proposition 21. Then π − ( A ) ∪ π − ( A ) = A , but neither of π − ( A i ) equals A , sincethe decomposition is proper, a contradiction.On the other hand, if A is not hollow, use the construction of Lemma 29. (cid:3) .2. Steady monoids. We say that a monoid (resp. monoid with zero element) S is left steady (resp. left -steady ) provided every compact left act in the category S − Act (resp. S − Act ) isnecessarily cyclic (cf. [4, 7]). Example 31. (1) If S is a group then indecomposable S -acts are cyclic. Hence compact S -actsare precisely cyclic ones by [11, Theorem I.5.10] (cf. Theorem 12). Thus groups are (left) steadymonoids.(2) The Pr¨ufer group Z p ∞ is a compact act over the monoid ( N , + , N , + , 0) is not steady. Proposition 32. Let C be either S − Act or S − Act . Then a projective left act is compact in C if and only if it is cyclic.Proof. The reverse implication is a consequence of Proposition 27.For the direct implication note that, since both S − Act and S − Act are UD-categories, byTheorem 15 any projective act has a decomposition into indecomposable projective subacts. As itis compact, it is indecomposable by Corollary 21. Now the result follows from Lemma 16 since S generates both the categories S − Act and S − Act . (cid:3) A monoid S is called left perfect ( left 0-perfect ) if every A ∈ S − Act ( A ∈ S − Act ) has aprojective cover, i.e. there exists (up to isomorphism unique) projective S -act P and an epimor-phism f : P → A such that for any proper subact P ′ ⊂ P the restriction f | P ′ : P ′ → A is not anepimorphism (cf. [8, 10]).. Proposition 33. Let S be a monoid with zero. If S is left -perfect, then compact objects of S − Act are precisely cyclic acts. Hence S is left -steady.Proof. By Proposition 27 it is enough to prove that a compact object of S − Act is necessarilycyclic. Let A be a compact S -act and π ∈ Mor( P, A ) be a projective cover of A . Assume that P isnot irreducible with a nontrivial decomposition ( P , P ). Then neither π ( P ) nor π ( P ) is not equalto A and B = π ( P ) ∩ π ( P ) is a subact of A . Then ( π ( P ) /B, π ( P ) /B ) forms a decompositionof Rees factor A/B . Note that it is non-trivial, otherwise π ( P ) ⊆ π ( P ) or π ( P ) ⊆ π ( P ) whichcontradicts to the fact that π ( P ) = A = π ( P ). Since every factor of A is indecomposable byLemma 25, we obtain a contradiction. (cid:3) Autocompact acts. Let us formulate the direct consequence of Lemma 23 and Proposi-tion 24: Lemma 34. Let C be an a autocompact object in either S − Act or S − Act and ϕ be an endomor-phism of C . Then ϕ ( C ) is autocompact and indecomposable, in particular, C is indecomposable. Theorem 35. The following conditions are equivalent for an act C ∈ S − Act : (1) C is autocompact, (2) C is compact, (3) C is indecomposable.Proof. The implication (2) ⇒ (1) is clear, the implication (1) ⇒ (3) follows from Lemma 34 and theequivalence (2) ⇔ (3) is proved in Lemma 25. (cid:3) Question. What about the situation in S − Act ? Example 36. Consider again the monoid Z = ( Z , · , 1) and the Z -act A = 2 Z ∪ Z from Ex-ample 31(1). Then A is auto-compact, since for any morphism A → A ` A with A i ∼ = A , thecomponent in which the image lies is determined by the image of the element 6.The previous example shows that within the category S − Act the class of autocompact actsis in general strictly larger than the class of compact acts; whereas the following example willshow that the class of autocompact acts is in general strictly smaller than that of indecomposableobjects, even for left perfect monoids. xample 37. Consider the monoid S = (cid:0)(cid:8) , , s, s (cid:9) , · , (cid:1) with the following multiplication table: · s s s s s s s s s . Then consider the S -act A = { x, y, z, t, θ } with the action of S given as follows: · a = θ for any a ∈ A · a = a for any a ∈ As · x = s · y = zs · z = ts · t = θ .A is indecomposable, while the Rees factor A/ h z i decomposes into two isomorphic components(so A is not compact), each of which can be mapped onto h t i ≤ A , hence A is not autocompact.One can furthermore prove that S is left perfect using [8, Thm 1.1]. Lemma 38. A non-hollow S -act A is a factor of a coproduct of a (suitable) pair of its propersubacts.Proof. Suppose A = B ∪ B with B i proper subacts of A . Then consider the following commutativediagram B ` B π −−−−→ A ν i x (cid:13)(cid:13)(cid:13) B i µ i −−−−→ A with ν i , µ i being the corresponding colimit injection and inclusion into A , respectively. Theuniversal property of colimit induces a morphism π : B ` B → A that is the desired epimor-phism. (cid:3) For S -acts A , A ∈ S − Act denote by π i : A ` A → A i , i = 1 , S − Act . Lemma 39. Let C, C , C ∈ S − Act and C ∼ = C ∼ = C . Then C is autocompact if and only iffor every morphism f : C → C ` C there exists i such that π i f ( C ) = θ .Proof. The direct implication follows immediately from Corollary 22.If C is not autocompact, then by Corollary 22 there exists morphism f : C → ` i ∈ I C i where C i ∼ = C and there exist i = j such that such that f ( C ) * ν i ( C i ) and f ( C ) * ν j ( C j ). Thus it isenough to compose f with the canonical projection to C i ` C j . (cid:3) For a pair B , B of subacts of a left S -act A with inclusions ι i B i → A denote by ρ B B : B ` B → A the unique morphism satisfying ρ B B ν i = ι i for i = 1 , S -acts in the category S − Act can by provided by nar-rowing the class of non-hollow (i. e. non-compact) acts by Proposition 40. The following conditions are equivalent for A, A , A ∈ S − Act such that A ∼ = A ∼ = A : (1) A is not autocompact in S − Act , (2) there exists a pair B , B of proper subacts of A satisfying A = B ∪ B and there exists amorphism f : B ` B → A ` A such that π i f = θ for i = 1 , and kerρ B B ⊆ ker f .Proof. Sufficiency follows from the Homomorphism Theorem [11, Theorem 4.21] which ensuresthe existence of a morphism f ′ : A → A ` A , which turns to be the witnessing morphism tonon-autocompactness thanks to the property π i f = θ for both i = 1 , ` B f −−−−→ A ` A ρ B B y x f ′ A A Necessity: Let g : A → A ` A be the morphism witnessing non-autocompactness, hence π i g = θ for both i = 1 , 2. Let ν i : A i → A ` A denote the colimit injection and set B i = g − ( g ( A ) ∩ ν i ( A i )); then obviously A = B ∪ B . Set now f = gρ B B . (cid:3) References [1] J. Ad´amek, H.Herrlich, G.E.Strecker, Abstract and Concrete Categories , John Wiley and Sons, Inc, 1990,http://katmat.math.uni-bremen.de/acc/acc.pdf.[2] D.M. Arnold, C.E. Murley, Abelian groups, A, such that Hom( A, − ) preserves direct sums of copies of A PacificJ. Math., 56 (1975), 7–20[3] H. Bass Algebraic K-theory , New York 1968, Benjamin.[4] R. Colpi and J. Trlifaj, Classes of generalized ∗ -modules , Comm. Algebra , 1994, 3985–3995.[5] A. Carboni, S. Lack, R.F.C.Walters, Introduction to extensive and distributive categories , Journal of Pure andApplied Algebra, 1993, , 145–158[6] J. Dvoˇr´ak: On products of self-small abelian groups, Stud. Univ. Babe-Bolyai Math. 60 (2015), no. 1, 13–17.[7] P.C. Eklof, K.R. Goodearl and J. Trlifaj, Dually slender modules and steady rings , Forum Math., 1997, ,61–74.[8] Isbell, J.: Perfect monoids. Semigroup Forum (1971) 2, 95–118.[9] P.K´alnai, J. ˇZemliˇcka, Compactness in abelian categories, J. Algebra, 534 (2019), 273–288.[10] Kilp, M.: Perfect monoids revisited. Semigroup Forum (1996) 53, 225–229.[11] M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, acts and categories, de Gruyter, Expositions in Mathematics29, Walter de Gruyter, Berlin 2000.[12] R. Khosravi, M. Roueentan, Co-uniform and hollow S -acts over monoids, arXiv: 1908.04559v1[13] R. Rentschler, Sur les modules M tels que Hom( M, − ) commute avec les sommes directes, C.R. Acad. Sci.Paris, , 1969, 930–933.[14] Roueentan, M., Sedaghatjoo, M.: On uniform acts over semigroups, Semigroup Forum (2018) 97, 229.[15] Sedaghatjoo, Mojtaba; Khaksari, Ahmad: Monoids over which products of indecomposable acts are indecom-posable, Hacet. J. Math. Stat. 46 (2017), No. 2, 229–237.[16] J. Trlifaj: Strong incompactness for some nonperfect rings, Proc. Amer. Math. Soc. 123 (1995), 21–25.[17] J. Trlifaj: Steady rings may contain large sets of orthogonal idempotents, in Abelian groups and objects(Padova, 1994) , Math. Appl., 343, Kluwer Acad. Publ., Dordrecht, 1995, 467–473.[18] J. ˇZemliˇcka and J. Trlifaj: Steady ideals and rings,