aa r X i v : . [ m a t h . C T ] J a n A new approach to S -protomodular categories Tamar Janelidze-GrayJanuary 27, 2021
Abstract
We propose a new approach to S -protomodular categories in the senseof D. Bourn, N. Martins-Ferreira, A. Montoli, and M. Sobral. Instead ofpoints (=split epimorphisms) it uses generalized points, which we defineas composable pairs of morphisms whose composites are pullback stableregular epimorphisms. This approach is convenient in describing the con-nection between split and regular Schreier epimorphisms of monoids. An S -protomodular category in the sense of D. Bourn, N. Martins-Ferreira, A.Montoli, and M. Sobral (see [3] and [5]) is a pointed category C (with finitelimits) equipped with a class of distinguished split epimorphisms (=“points”)satisfying certain conditions; we will recall the precise definition in Section 3.The introduction to [5] says (among many other things):“Note that our approach to relative non-abelian homological algebra is dif-ferent from the one initiated by T. Janelidze in [7] and developed by her inseveral later papers: in our work, the word “relative” refers to a chosen class ofpoints, i.e. of split epimorphisms with specified splitting, while in T. Janelidze’spapers it refers to a chosen class of (not necessarily split) regular epimorphisms.”Following this remark, in a sense, in this paper we develop what mightbe called non-split (epimorphism) approach to the theory of S -protomodularcategories . We replace points ( f, s ) = A f / / B, f s = 1 Bs o o with pairs ( f, g ) = C g / / A f / / B where the composite f g is required to be a pullback stable regular epimorphism,which we call generalized points . We impose certain conditions on a class T of generalized points in a category C , and call C a T -protomodular category C with finite limits, a bijection between thecollections of classes S making C an S -protomodular category and the collectionsof classes T making C a T -protomodular category. In this formulation we havein mind of course that S always denotes a class of points while T always denotesa class of generalized points.In the last section we take C to be the category of monoids and proveTheorems 4.5 and 4.6, which show that using generalized points is convenientin describing the connection between split and regular Schreier epimorphismsof monoids.A remark on terminology: The terms strong (point) and regular (Schreierepimorphism) we are using might sound confusing since there more standardand familiar strong epimorphisms and regular epimorphisms in general categorytheory. But this choice of terminology was already made several years ago inthe literature we refer to. Throughout this paper we assume that C is a pointed category with finite limits. Definition 1.
A generalized point in C is a pair ( f, g ) , in which f : A → B and g : C → A are morphisms in C such that the composite f g : C → B is apullback stable regular epimorphism in C . We will also write C g / / A f / / B (2.1)for the generalized point ( f, g ) in C . Note that since f g is a pullback stable reg-ular epimorphism, it follows that f is also a pullback stable regular epimorphism(see Proposition 1.5 of [6]).A morphism between two such generalized points ( f, g ) and ( f ′ , g ′ ) is a triple( α, β, γ ), in which α : A → A ′ , β : B → B ′ , and γ : C → C ′ are morphisms in C such that the diagram C g / / γ (cid:15) (cid:15) A α (cid:15) (cid:15) f / / B β (cid:15) (cid:15) C ′ g ′ / / A ′ f ′ / / B ′ (2.2)commutes.We will denote by GPt ( C ) the category of all generalized points in C . Notethat if we take all those generalized points (2.1) for which the composite f g isthe identity, then we obtain the category of points Pt ( C ) in C in the sense ofD. Bourn [1]. We will also call such generalized points split generalized points. efinition 2. A generalized point ( f, g ) is said to be a strong generalized point,if g and the kernel of f are jointly strongly epic. In particular, a split generalized point that is strong is the same as a strongsplit epimorphism in the sense of [2], and the same as a regular point in thesense of [8], and a strong point in the sense of [5].Let ( f, g ) be a generalized point in C and let x : X → B be any morphismin C . Taking the pullbacks ( A × B X, π , π ) of f and x , and ( C × B X, π ′ , π ′ )of f g and x , we obtain the commutative diagram C × B X g × / / π ′ (cid:15) (cid:15) A × B X π (cid:15) (cid:15) π / / X x (cid:15) (cid:15) C g / / A f / / B (2.3)in which π ( g ×
1) = π ′ , and not only the second square but also the first one isa pullback. Since ( f, g ) is a generalized point, so is also the pair ( π , g × π , g ×
1) is the pullback of ( f, g ) along x in GPt ( C ).We have: Theorem 1.
If in the commutative diagram (2.3), ( f, g ) is a generalized pointin C , x is a pullback stable regular epimorphism, and ker( π ) and g × arejointly strongly epic, then ker( f ) and g are also jointly strongly epic.Proof. Let m : M → A be a monomorphism, and let u : Ker( f ) → M and v : C → M be any morphisms with mu = ker( f ) and mv = g . We have toprove that, under the assumptions of the theorem, m is an isomorphism. For,consider the diagram C × B X × v % % g × / / π ′ (cid:15) (cid:15) A × B X π / / π (cid:15) (cid:15) X x (cid:15) (cid:15) M × B X m × ♠♠♠♠♠♠♠♠♠♠♠♠ π ′′ (cid:15) (cid:15) K u ′ b b k ′ F F ☞☞☞☞☞☞☞☞☞☞☞☞☞☞ u | | ②②②②②②②② k (cid:25) (cid:25) ✷✷✷✷✷✷✷✷✷✷✷✷✷✷ M m ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ C v ssssssssss g / / A f / / B (2.4)in C , which adds more arrows to diagram (2.3) as follows: • Since the right-hand square of (2.4) is a pullback, we can identify Ker( π )with Ker( f ), and this object is denoted by K ; accordingly, we can write( K, k ) = (Ker( f ) , ker( f )) and ( K, k ′ ) = (Ker( π ) , Ker( π )).3 m , u , and v are as above. • M × B X is obtained as the pullback of f m and x ; its first pullback projec-tion is denoted by π ′′ , while the second one coincides with the compositeof m × M × B X → A × B X and π : A × B X → X . It follows that( M × B X, π ′′ , m ×
1) is the pullback of m and π , therefore m × m . Note also that π is a pullback stable regularepimorphism since so is x . • Since mu = k = π k ′ , there exists a (unique) morphism u ′ : K → M × B X such that ( m × u ′ = k ′ (and also π ′′ u ′ = u ).Since k ′ = ker( π ) and g × m × π is a pullback stable regular epimorphismand ( M × B X, π ′′ , m ×
1) is the pullback of m and π , it follows that m is anisomorphism (see e.g. Proposition 1.6 of [6]).Note the following simple fact: Proposition 1.
In a commutative diagram B s (cid:15) (cid:15) h (cid:127) (cid:127) ⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦ B (cid:31) (cid:31) ❅❅❅❅❅❅❅❅❅❅❅ C g / / A f / / B in C , if ker( f ) and s are jointly strongly epic, then ( f, g ) is a generalized strongpoint in C . Given a generalized point ( f, g ) as in (2.3), consider the commutative dia-gram C h C , C i y y tttttttttttttt h g, C i (cid:15) (cid:15) C " " ❋❋❋❋❋❋❋❋❋❋❋❋❋ C × B C g × / / π ′ (cid:15) (cid:15) A × B C π (cid:15) (cid:15) π / / C fg (cid:15) (cid:15) C g / / A f / / B whose bottom part is the same as diagram (2.3) with x = f g . From Theorem 1and Proposition 1, we obtain: Corollary 1.
In the notation above, if h g, C i and ker( π ) are jointly stronglyepic, then ( f, g ) is a generalized strong point. T-protomodular categories
Let S and T be classes of points (= split generalized points) and of generalizedpoints, respectively, in C . Consider the following conditions: Condition 1.
The class S has the following properties:(a) it is pullback stable;(b) it is closed under finite limits in Pt ( C ) ;(c) all its elements are strong. Condition 2.
The class T has the following properties:(a) it is pullback stable;(b) it is closed under component-wise finite limits in GPt ( C ) ;(c) all its elements are strong;(d) in a diagram of the form C h g, C i / / A × B C π (cid:15) (cid:15) π / / C fg (cid:15) (cid:15) C g / / A f / / B, where ( f, g ) is a generalized point and the right-hand square is a pullback,the top row belongs to T if and only if the bottom row does. According to [5] (originally from [3]), to say that C is S - protomodular is tosay that S satisfies Condition 1, and we introduce: Definition 3.
We say that C is T -protomodular if T satisfies Condition 2. The purpose of this section is to prove the following theorem, which in factsays that the notions of S -protomodular and of T -protomodular are equivalentto each other: Theorem 2.
Let S and T be the collections of all classes of points and of allclasses of generalized points, respectively, in C . Let F : T → S and G : S → T be the maps defined by F ( T ) = T ∩ Pt ( C ) , G ( S ) = { ( f, g ) ∈ GPt ( C ) | ( π , h g, C i ) ∈ S } , where π and h g, C i are as in the diagram used in Condition 3.2(d). Then F and G induce inverse to each other bijections between the collection of all T ∈ T ,such that C is a T -protomodular category, and the collection of all S ∈ S , suchthat C is an S -protomodular category. roof. Suppose S ∈ S and T ∈ T satisfy Conditions 3.1 and 3.2, respectively;we have to verify that:(i) F ( T ) satisfies Condition 3.1;(ii) G ( S ) satisfies Condition 3.2;(iii) GF ( T ) = T ;(iv) F G ( S ) = S .Let us give details of these verifications:(i) is obvious, having in mind that all finite limits in Pt ( C ) are component-wise.(ii), Condition 1(a): Given ( f : A → B, g : C → A ) ∈ G ( S ) and a morphism x : X → B , consider the commutative diagram C × B X } } ④④④④④④④ / / A × B C × B X x x ♣♣♣♣♣♣♣♣♣♣ (cid:15) (cid:15) / / C × B X } } ④④④④④④④ (cid:15) (cid:15) C / / A × B C (cid:15) (cid:15) / / C (cid:15) (cid:15) C × B X } } ④④④④④④④ / / A × B X w w ♣♣♣♣♣♣♣♣♣♣♣ / / X x } } ③③③③③③③③ C g / / A f / / B (with obviously defined unlabeled arrows). In this diagram, since each face ofthe right-hand cube is a pullback diagram, we have: • since ( f, g ) belongs to G ( S ), the second row belongs to S ; • since the second row belongs to S , and S is pullback stable, the first rowalso belongs to S ; • since the first row belongs to S , the third row, which is nothing but thepullback ( f, g ) along x , belongs to G ( S ).(ii), Condition 3.2(b): Consider the extension D of GPt ( C ) defined simplyas the category of all composable pairs of morphisms in C . Define the forgetfulfunctor U : D → C by U ( f : A → B, g : C → A ) = B , and, for an object B in C , write ( D ↓ B ) for the fibre of this functor over B . G ( S ) does satisfyCondition 3.2(b) since: • the component-wise limits in GPt ( C ) are calculated as in D ; • G ( S ) (obviously) contains the terminal object (0 → , →
0) of D ; • for every object B , the fibre inclusion functor ( D ↓ B ) → D preservespullbacks, and the same is true for Pt ( C );6 for every morphism x : X → B , the change-of-base functor x ∗ : ( D ↓ B ) → ( D ↓ X ) preserves all finite limits, and, again, the same is true for Pt ( C ).(ii), Condition 3.2(c): G ( S ) does satisfy it by Corollary 1.(ii), Condition 3.2(d): G ( S ) does satisfy it since (obviously) the top row ofthe diagram belongs to G ( S ) if and only if it belongs to S .(iii): For a generalized point ( f, g ), we have( f, g ) ∈ GF ( T ) ⇔ ( π , h g, C i ) ∈ F ( T ) ⇔ ( π , h g, C i ) ∈ T ⇔ ( f, g ) ∈ T , where the second equivalence holds because ( π , h g, C i ) automatically belongsto Pt ( C ), while the third one holds by Condition 2(d).(iv): For a point ( f, g ), we have( f, g ) ∈ F G ( S ) ⇔ ( f, g ) ∈ G ( S ) ⇔ ( f, g ) ∈ S , since, when ( f, g ) is a point, it is the same as ( π , h g, C i ). Throughout this section we assume that C is a category of monoids, for whichwe will use the additive notation. Since it is a variety of universal algebras, ageneralized point in C is a pair ( f, g ) for which f g is a surjective homomorphism(of monoids). Let us recall two known definitions (see e.g. [10], [8], [4], [3], [9]): Definition 4.
A point (=split generalized point) ( f, g ) = B g / / A f / / B (in C ) is said to be a Schreier point, if for every a ∈ A there exists a unique k ∈ Ker( f ) such that a = k + gf ( a ) . Definition 5.
Let f : A → B be a surjective homomorphism of monoids. Then:(a) an element a in A is said to be a representative of an element b in B ,if f ( a ) = b and for every a ′ ∈ A with f ( a ′ ) = b , there exists a uniqueelement k ∈ Ker( f ) with a ′ = k + a ;(b) f is said to be a Schreier epimorphism if every element of B has a repre-sentative;(c) f is said to be a regular Schreier epimorphism if it is a Schreier epimor-phism and the set of all representatives of elements of B is a submonoidof A . Let us introduce: 7 efinition 6.
A generalized point ( f, g ) = C g / / A f / / B (in C ) is said to be a Schreier generalized point, if for every a ∈ A and every c ∈ C with f ( a ) = f g ( c ) , there exists a unique k ∈ Ker( f ) such that a = k + g ( c ) . Remark 1.
Every Schreier generalized point is obviously strong.
The following two theorems describe the connection between Definitions 4,5(c), and 6.
Theorem 3.
Let S and T be the classes of Schreier points and of Schreiergeneralized points, respectively. Then:(a) C is an S -protomodular cateogry in the sense of [5] (this is proved in [3]);(b) C is a T -protomodular category in the sense of our Definition 3;(c) S and T correspond to each other under the bijection described in Theorem2.Proof. As follows from 3( a ) (which was poved in [3]) and Theorem 2, all we needto show is that G ( S ) = T . That is, we only need to show that, in the notationabove, ( f, g ) is a Schreier generalized point if and only if ( π , h g, C i ) = C h g, C i / / A × B C π / / C is a Schreier point. However, this is straighforward, having in mind that: • ( a, c ) ∈ A × C belongs to A × B C if and only if a ∈ A and c ∈ C are suchthat f ( a ) = f g ( c ); • Ker( π ) = Ker( f ) × { } ; • ( a, c ) = ( k,
0) + h g, C i π ( a, c ) ⇐⇒ ( a, c ) = ( k,
0) + ( g ( c ) , c ) ⇐⇒ a = k + g ( c ). Theorem 4.
A monoid homomorphism f : A → B is a regular Schreier epi-morphism if and only if there exists a monoid homomorphism g : C → A making ( f, g ) a Schreier generalized point.Proof. “If”: Suppose ( f, g ) is a Schreier generalized point. Since f g is surjective,for every b ∈ B there exists c ∈ C with f g ( c ) = b , making g ( c ) a representativeof b . Therefore, f is a Schreier epimorphism. It remains to prove that if a and a ′ are representatives of b and b ′ respectively, then a + a ′ is a representative (of b + b ′ ). As follows from Proposition 3.5 of [9], we can replace a with any otherrepresentative b , and a ′ with any other representative of b ′ . We replace a with8 ( c ), and a ′ with g ( c ′ ), where f g ( c ) = b and f g ( c ′ ) = b ′ , and we know that g ( c ) + g ( c ′ ) = g ( c + c ′ ) is a representative (of b + b ′ ).“Only if”: Suppose f : A → B is a regular Schreier epimorphism. Let C bethe free monoid on B , and η : B → C the canonical inclusion map. For each b ∈ B choose any representative a b of b , and define g : C → B as the uniquemonoid homomorphism carrying η ( b ) to a b , for each b ∈ B . To prove that ( f, g )is a Schreier generalized point it suffices to prove that each g ( c ) ( c ∈ C ) is arepresentative, but this follows from the fact that so are all g ( η ( b )) = a b ( b ∈ B )and C is generated by η ( B ). Corollary 2.