aa r X i v : . [ m a t h . C T ] N ov Anticommutativity and the triangular lemma.
Michael Hoefnagel
Abstract
For a variety V , it has been recently shown that binary products com-mute with arbitrary coequalizers locally, i.e., in every fibre of the fibrationof points π : Pt( C ) → C , if and only if Gumm’s shifting lemma holds onpullbacks in V . In this paper, we establish a similar result connectingthe so-called triangular lemma in universal algebra with a certain cat-egorical anticommutativity condition. In particular, we show that thisanticommutativity and its local version are Mal’tsev conditions, the localversion being equivalent to the triangular lemma on pullbacks. As a corol-lary, every locally anticommutative variety V has directly decomposablecongruence classes in the sense of Duda, and the converse holds if V isidempotent. Recall that a category is said to be pointed if it admits a zero object
0, i.e.,an object which is both initial and terminal. For a variety V , being pointed isequivalent to the requirement that the theory of V admit a unique constant.Between any two objects X and Y in a pointed category, there exists a uniquemorphism 0 X,Y from X to Y which factors through the zero object. The pres-ence of these zero morphisms allows for a natural notion of kernel or cokernel ofa morphism f : X → Y , namely, as an equalizer or coequalizer of f and 0 X,Y ,respectively. Every kernel/cokernel is a monomorphism/epimorphism, and amonomorphism/epimorphism is called normal if it is a kernel/cokernel of somemorphism. Given any pointed category C with binary products, the productinclusion X ι −→ X × Y (i.e., the unique morphism into X × Y determined by1 X and 0 X,Y ) is always a kernel of the product projection X × Y π −→ Y , butit is not generally true that π is a cokernel of ι . Pointed categories whereevery product projection is normal are said to have normal projections [25]. Forexample, every subtractive [27] or unital category [6] has normal projections. Inparticular, every pointed subtractive variety or any J´onsson-Tarski variety hasnormal projections.To any object X , in any category C , we may associate the pointed categoryPt C ( X ) of split epimorphisms with codomain X and chosen splitting — theso-called “category of points” of X . Explicitly, objects of this category aretriples ( A, p, s ) where A is an object of C and p : A → X and s : X → A are morphisms in C with p ◦ s = 1 X . A morphism f : ( A, p, s ) → ( B, q, t ) in1t C ( X ) is a morphism f : A → B in C such that q ◦ f = p and f ◦ s = t , i.e.,it is a morphism which makes the upward and downward facing triangles in thediagram below commute. A f / / p % % B q y y X t : : s e e The categories Pt C ( X ) arise as the fibres of a certain functor π : Pt( C ) → C ,namely, the fibration of points [6]. This functor plays an important role in cate-gorical algebra, as it classifies many notions of central importance to the subjectsuch as, for example, the notion of a Mal’tsev category [2, 1]: a finitely completecategory C is Mal’tsev — i.e., C satisfies the categorical condition associatedwith Mal’tsev varieties, if and only if Pt C ( X ) is unital [6] — i.e., Pt C ( X ) sat-isfies the categorical condition associated with J´onsson-Tarski varieties. Thereis a canonical zero object in Pt C ( X ) which is given by ( X, X , X ), and withrespect to this zero object the zero morphism from ( A, p, s ) to (
B, q, t ) is simplythe composite t ◦ p . If C has pullbacks, then we may form products in Pt C ( X )in the following way. Consider a pullback A × X B of p along q : X ( s,t ) t " " s % % A × X B p / / p (cid:15) (cid:15) d ❍❍❍❍ ❍❍❍❍❍ B q (cid:15) (cid:15) A p / / X Since p ◦ s = 1 X = q ◦ t , the universal property of the pullback determines aunique morphism ( s, t ) with the property that p ◦ ( s, t ) = s and p ◦ ( s, t ) = t .Then it is readily checked that the diagram( A, p, s ) p ←− ( A × X B, d, ( s, t )) p −→ ( B, q, t ) , is a product diagram in Pt C ( X ). Now, a finitely complete category C is thensaid to have normal local projections [26] if for every object X in C , the cat-egory Pt C ( X ) has normal projections. For varieties of algebras, it has beenrecently shown in [18] that normal local projections is closely related to the(local) commutativity of binary products with arbitrary coequalizers, which inturn is closely related to Gumm’s shifting lemma [15]. Recall that an algebra X satisfies the shifting lemma if for any congruences R, S, T on X such that R ∩ S T and xRu, yRv and xSy, uSv , then uT v implies xT y . A varietyis said to satisfy the shifting lemma if every algebra in it satisfies the shift-ing lemma, which happens if and only if the variety is congruence modular.The implications of relations in the shifting lemma are usually depicted with a2iagram: u TSR v R x TS y Given morphisms p : A → X and q : B → X in a variety of algebras V thentheir pullback is given by A × X B = { ( a, b ) | p ( a ) = q ( b ) } together with thecanonical projection morphisms p : ( a, b ) a and p : ( a, b ) b . The resultalluded to earlier connecting normal local projections with the shifting lemma,is that a variety V has normal local projections if and only if V satisfies theshifting lemma on pullbacks : for any morphisms p : A → X and q : B → X in V and any congruence Θ on A × X B we have ( x, u )Θ( y, u ) ⇒ ( x, v )Θ( y, v )for any elements ( x, u ) , ( y, u ) , ( x, v ) , ( y, v ) of A × X B . This can be seen as theshifting lemma in the special case where R = Eq( p ) and S = Eq( p ) are thekernel congruence relations of the morphisms p and p , respectively.( x, u ) ΘEq( p )Eq( p ) ( y, u ) Eq( p ) ( x, v ) ΘEq( p ) ( y, v )This is because we always have Eq( p ) ∩ Eq( p ) ⊆ Θ. Moreover, a variety V satisfies the shifting lemma on pullbacks if and only if finite products commutewith arbitrary coequalizers in Pt V ( X ) for any algebra X in V (see Theorem 3.7in [18]). Recall that an algebra X in a variety is said to satisfy the triangularlemma [13] (see also [11]) if for any three congruences R, S, T on X such that R ∩ S T , if xRySz and xT z , then yT z . z S x T ⑧⑧⑧⑧⑧⑧⑧⑧ R y T The main aim of this paper is to illustrate a similar result connecting the trian-gular lemma with a natural anticommutativity condition (Definition 2.1) whichis based the notion of commuting morphisms is the sense of Huq [22] (see Def-inition 2.1 below). In particular, we show that for varieties, this notion ofanticommutativity, as well as its local version, are both Mal’tsev properties, the3atter being equivalent to the triangular lemma restricted to pullbacks: a variety V is locally anticommutative if and only if for any two morphisms p : A → X and q : B → X , and any congruence Θ on the pullback A × X B of p along q , if( x, y ) , ( x ′ , y ) , ( x ′ , y ′ ) ∈ A × X B are any elements such that ( x, y )Θ( x ′ , y ′ ), then( x ′ , y )Θ( x ′ , y ′ ). ( x ′ , y ′ ) Eq( p ) Θ ( x, y ) Θ ✈✈✈✈✈✈✈✈✈ Eq( p ) ( x ′ , y ) Internal relations in categories
Given any two objects X and Y in a category C with binary products, recallthat an internal binary relation R from X to Y is simply a monomorphism r =( r , r ) : R → X × Y , and the morphisms r and r are called the projections of the relation. In what follows, we will say that r represents the relation R .Throughout the rest of this paper, we will adhere to the following notation:given morphisms x : S → X and y : S → Y in C we write xRy or ( x, y ) ∈ S R if the morphism ( x, y ) : S → X × Y factors through r , i.e., there exists a(necessarily unique) morphism u : S → R such that r ◦ u = ( x, y ). Thisnotation extends to relations of higher arity: if R is an internal n -ary relation r : R → X × · · · × X n , then for any morphisms x i : S → X i for i = 1 , . . . , n ,we write ( x , x , . . . , x n ) ∈ S R if the morphism ( x , x , . . . , x n ) : S → X ×· · · × X n factors through r . For two internal relations R, R ′ between objects X , ..., X n , we write R R ′ if the representing monomorphism of R factorsthrough the representing monomorphism of R ′ . With respect to this notation,we can reformulate familiar properties of relations internal to categories. Forexample an internal binary relation R on an object X in C is transitive if forany x, y, z : S → X we have xRy and yRz implies xRz . In a similar way we canreformulate the property of a binary relation to be reflexive, symmetric or anequivalence, internal to categories. Moreover, we can also reformulate variousshifting properties, such as the triangular lemma, in general categories (seeDefinition 1.2 below). If C has pullbacks, then the intersection of two relationscan be defined. For example, if R and T are binary relations represented bythe monomorphisms r = ( r , r ) : R → X × Y and t = ( t , t ) : T → X × Y respectively, then their intersection R ∩ T is represented by the diagonalmorphism in any pullback ( R ∩ T ) / / (cid:15) (cid:15) % % ❑❑❑❑❑❑❑❑❑❑ T t (cid:15) (cid:15) R r / / X × Y since the diagonal morphism in a pullback as above is always a monomorphism.This relation is then easily seen to posses the property ( x, y ) ∈ S R ∩ T if and4nly if ( x, y ) ∈ S R and ( x, y ) ∈ S T for any x : S → X and y : S → Y . Remark 1.1.
This technique of working “set-theoretically” within abstractcategories is the standard one of working with “generalized elements”, which isformally validated by the Yoneda embedding Y : C → Set C op . Definition 1.2.
Let C be a category with binary products. An object X in acategory C satisfies the triangular lemma if for any three (internal) equivalencerelations R, S, T on the same object X such that R ∩ S T , if xRySz and xT z ,then yT z , for any morphisms x, y, z : S → X . Then C is said to satisfy thetriangular lemma if every object in C does.Given a variety V , by making use of the free algebra over one generator in V itis an easy exercise to show that V satisfies the triangular lemma in the senseof Definition 1.2 if and only if it satisfies the triangular lemma in the univer-sal algebraic sense. Any congruence distributive variety satisfies the triangularlemma, but not every variety satisfying the triangular lemma is congruence dis-tributive as shown in [16]. However, a congruence modular variety is congruencedistributive if and only if it satisfies the triangular lemma [23]. But also, anyregular majority category in the sense of [17] satisfies the triangular lemma inthe sense of Definition 1.2 (see Lemma 1.1 and the paragraph preceding it in[28]). Thus for example, the dual category Top op is a regular majority category(Example 2.6 in [17]), and hence it satisfies the triangular lemma. Convention
We will sometimes denote zero morphisms between objects ina pointed category simply by 0, leaving out the domain and codomainreference as subscripts.
Recall that two morphisms f : A → C and g : B → C in a pointed category C with binary products are said to commute [22] (or “cooperate” [8]) if thereexists a morphism ρ : A × B → C such that ρ ◦ ι = f and ρ ◦ ι = g , where ι : A → A × B and ι : B → A × B are the canonical product inclusions.Following the terminology of [8], we will call such a morphism ρ a cooperator for f and g in what follows. A f ❋❋❋❋❋❋❋❋❋ ι / / A × B ρ (cid:15) (cid:15) B ι o o g { { ①①①①①①①①① C A morphism f : X → Y in C is called central when it commutes with the identity1 Y on Y , and an object M is called commutative if 1 M is central. For example, inthe category Grp of groups, two morphisms f : G → L and g : H → L commuteif and only the subgroups f ( G ) and g ( H ) commute in the usual sense. Thecategory Imp of (non-empty) implications algebras [29] is pointed, and if two5orphisms f : A → C and g : B → C commute then f ( A ) ∩ g ( B ) = 0. In otherwords, if two morphisms f and g of implication algebras commute, then theyare disjoint in the following sense: two morphisms f : X → Z and g : Y → Z ina pointed category C are said to be disjoint if for any commutative diagram S x / / y (cid:15) (cid:15) Y g (cid:15) (cid:15) X f / / Z we have g ◦ x = 0 = f ◦ y . This brings us to the main definition of this paper: Definition 2.1.
A pointed category C with binary products is a called anti-commutative if every pair of commuting morphisms are disjoint. Proposition 2.2.
Let C be a pointed category with binary products and kernels,and suppose that f : X → Z and g : Y → Z are any pair of morphisms in C .Then f and g are disjoint if and only if the diagram ker( f ) × ker( g ) p (cid:15) (cid:15) p / / Y g (cid:15) (cid:15) X f / / Z is a pullback, where p and p are the canonical product projections composedwith the canonical kernel inclusions.Proof. If f and g are disjoint, and x : S → X and y : S → Y are any morphismssuch that f ◦ x = g ◦ y , then anticommutativity implies that f ◦ x = 0 = g ◦ y so that x and y factor through ker( f ) and ker( g ) respectively, and hence ( x, y )factors through ker( f ) × ker( g ). Conversely, if the diagram above is a pullback,then f ◦ x = g ◦ y gives a factorization of both x and y through ker( f ) and ker( g )respectively, so that f ◦ x = 0 = g ◦ y . Remark 2.3.
The content of the above proposition was essentially what wasused in [17] in order to show that every pointed finitely complete majoritycategory is anticommutative (see Proposition 3.8).Recall that a category C is said to be regular [3] it satisfies the following threeproperties:(i) C has all finite limits.(ii) C has coequalizers of kernel-pairs.(iii) The class of all regular epimorphisms in C is pullback stable, i.e., for anypullback diagram • / / p (cid:15) (cid:15) • e (cid:15) (cid:15) • / / • e is a regular epimorphism, then so is p .For example, any (quasi)variety of algebras is a regular category. Every mor-phism f : X → Y in a regular category C admits a factorization f = m ◦ e where e : X → I is a regular epimorphism and m is a monomorphism. Sucha factorization of f is called an image-factorization of f and is unique up tounique isomorphism. If C is a variety of algebras, then the usual projection X → f ( X ) followed by the inclusion f ( X ) → Y yields an image factorizationfor f . In what follows, we will write X e f −→ f ( X ) m f −−→ Y for a chosen imagefactorization of a morphism f : X → Y in a regular category. Proposition 2.4.
For any pointed regular category C and any morphisms f : X → Z and g : Y → Z in C , f and g are disjoint if and only if m f and m g are disjoint. Note that according to Proposition 2.2, two monomorphisms in a pointed cate-gory with binary products and kernels are disjoint if and only if their pullbackis a zero-object.
Proof.
Consider the diagram below where the bottom row and right column areimage factorizations of f and g respectively and each square is a pullback. P e " " " " ❊❊❊❊❊❊❊❊❊❊ p / / / / p (cid:15) (cid:15) (cid:15) (cid:15) K (cid:15) (cid:15) (cid:15) (cid:15) k / / Y e g (cid:15) (cid:15) (cid:15) (cid:15) K k (cid:15) (cid:15) / / / / I m ❋❋❋❋❋❋❋❋❋❋❋ (cid:15) (cid:15) / / g ( Y ) m g (cid:15) (cid:15) X e f / / / / f ( X ) m f / / Z In the above diagram the diagonal morphism e is a regular epimorphism (as itis a composite of regular epimorphisms in a regular category) and the morphism m is a monomorphism. Therefore, if f and g are disjoint then m ◦ e = 0 whichimplies that m = 0 so that m f and m g are disjoint. On the other hand, if m f and m g are disjoint then I is a zero object, and hence each square being a pullbackimplies that K = ker( f ) and K = ker( g ) and that P = ker( f ) × ker( g ), p and p are the canonical product projections, k and k are the kernel inclusions, sothat f and g are disjoint by Proposition 2.2. Proposition 2.5.
For a pointed category C with binary products, we have ( i ) = ⇒ ( ii ) = ⇒ ( iii ) where:(i) C is anticommutative.(ii) Every central morphism in C is a zero morphism.(iii) Every commutative object in C is a zero object. roof. For ( i ) = ⇒ ( ii ), note that if f : X → Y is central, then it commuteswith the identity 1 Y , so that f and 1 Y are disjoint, so that f = 0. Note that( ii ) = ⇒ ( iii ) is trivial.A pointed finitely complete category C is called unital [6] if for any commutativediagram • m (cid:15) (cid:15) A ι / / ; ; ✇✇✇✇✇✇✇✇✇ A × B B ι o o c c ●●●●●●●●● where ι and ι are the canonical product inclusions, if m is a monomorphismthen m is an isomorphism. This is equivalent to requirement that the productinclusions ι and ι are jointly strongly epimorphic . A variety is then unital inthe above sense if and only if it is a J´onsson-Tarski variety. The corollary belowshows that the converse implications in the statement of Proposition 2.5 holdwhen the base category is regular and unital. Corollary 2.6.
For a regular unital category C the following are equivalent:(i) C is anticommutative.(ii) Every central morphism in C is .(iii) Every commutative object in C is trivial.Proof. The implications ( i ) = ⇒ ( ii ) = ⇒ ( iii ) follow from Corollary 2.5. For( iii ) = ⇒ ( i ), we note that if f : X → Z and g : Y → Z are two morphisms whichcommute, then C being unital, their images f ( X ) and g ( Y ) commute. Theirintersection f ( X ) ∩ g ( Y ) is a commutative object in C (see Example 1.4.2 in [5]),and hence, f ( X ) ∩ g ( Y ) = 0 so that f and g are disjoint by Proposition 2.4.If C is not regular, then ( ii ) need not imply ( i ). For example, consider the fullsubcategory CGrp of Grp consisting centerless groups , i.e., groups G for whichZ( G ) = { x ∈ G | ∀ y ∈ G ( xy = yx ) } = 0. The category CGrp has products,is unital (product inclusions are jointly strongly epimorphic) and every centralmorphism in
CGrp is a zero morphism, however it is not anticommutative. Tosee this, consider the free group F over two generators x, y , and consider themorphism f : F → F such that f ( x ) = x = f ( y ). Note that F is centerless, sothat f is a morphism in CGrp . Moreover, f commutes with itself, since themap ρ : F → F defined by ( a, b ) f ( a ) · f ( b ) is a cooperator for f with itself,but f is not disjoint with itself. Proposition 2.7.
For a pointed category C with binary products the followingare equivalent.(i) For any morphism ρ : X × X → Y if ρ ◦ ι = ρ ◦ ι then ρ ◦ ι = 0 = ρ ◦ ι .(ii) For any morphism ρ : X × X → Y we have ρ ◦ ( x,
0) = ρ ◦ (0 , x ) = ⇒ ρ ◦ ( x,
0) = 0 = ρ ◦ (0 , x ) for any morphism x : S → X . iii) For any morphism ρ : X × Y → Z we have ρ ◦ ( x,
0) = ρ ◦ (0 , y ) = ⇒ ρ ◦ ( x,
0) = 0 = ρ ◦ (0 , y ) for any morphisms x : S → X and y : S → Y .(iv) C is anticommutative.Proof. For ( i ) = ⇒ ( ii ), let x : S → X be any morphism such that ρ ◦ ( x,
0) = ρ ◦ (0 , x ). Consider the composite morphism S × S x × x −−−→ X × X ρ −→ Y . Then wehave that ( ρ ◦ ( x × x )) ◦ ι = ρ ◦ ( x,
0) = ρ ◦ (0 , x ) = ( ρ ◦ ( x × x )) ◦ ι , whichimplies that: ρ ◦ ( x,
0) = ( ρ ◦ ( x × x )) ◦ ι = 0 = ( ρ ◦ ( x × x )) ◦ ι = ρ ◦ (0 , x ) . For ( ii ) = ⇒ ( iii ) suppose that x : S → X and y : S → Y are any morphismssuch that ρ ◦ ( x,
0) = ρ ◦ (0 , y ). Consider the morphism ( X × Y ) × ( X × Y ) p −→ Z × Z which is defined by the map (( a, b ) , ( c, d )) ( ρ ( a, d ) , ρ ( c, b )). Then we havethat p ◦ (( x, y ) , (0 , ρ ◦ ( x, , ρ ◦ (0 , y ))= ( ρ ◦ (0 , y ) , ρ ◦ ( x, p ◦ ((0 , , ( x, y )) , which implies p ◦ (( x, y ) , (0 , ρ ◦ ( x, , ρ ◦ (0 , y )) = (0 ,
0) and the resultfollows. For ( iii ) = ⇒ ( iv ) suppose that ρ is a cooperator for two morphisms f : X → Z and g : Y → Z , and let x : S → X and y : S → Y be any twomorphisms such that f ◦ x = g ◦ y . Consider the diagram: X ι (cid:15) (cid:15) f ❋❋❋❋❋❋❋❋❋ S x ; ; ①①①①①①①①① y ❋❋❋❋❋❋❋❋❋ X × Y ρ / / ZY ι O O g ; ; ①①①①①①①①① The commutativity of outer square gives ρ ◦ ( x,
0) = f ◦ x = g ◦ y = ρ ◦ (0 , y ) = ⇒ f ◦ x = 0 = g ◦ y. For ( iv ) = ⇒ ( i ), just note that ρ is a cooperator for ρ ◦ ι and ρ ◦ ι The following proposition is a simple reformulation of ( i ) in the above proposi-tion when C has coequalizers. Proposition 2.8.
A pointed category C with binary products and coequalizersis anticommutative if and only if for any object X in C we have q ◦ ι = 0 = q ◦ ι for any coequalizer q : X × X → Q of ι and ι . .1 Examples of anticommutative categories Given any morphism f : X → Y in a finitely complete category, the kernelequivalence relation Eq( f ) of f is the equivalence relation obtained by pullingback f along itself. If C is a variety of algebras, then Eq( f ) is the kernelcongruence associated to f , i.e., we have:Eq( f ) = { ( x, y ) ∈ X × X | f ( x ) = f ( y ) } . Example 2.1. If V is a pointed variety of universal algebras which admits anidempotent binary operation b ( x, y ) satisfying b ( x,
0) = 0 = b (0 , y ), then V sat-isfies (ii) of Proposition 2.7. Since for any homomorphism f : X × X → Y in V if f ( x,
0) = f (0 , x ) then f ( x,
0) = b ( f ( x, , f ( x, b ( f ( x, , f (0 , x )) = f ( b ( x, , b (0 , x )) = 0. Thus, the variety of meet-semilattices with a least ele-ment is anticommuative. In Theorem 3.1 below, a Mal’tsev condition is givenfor a pointed variety to be anticommutative.A variety V is said to have directly decomposable congruence classes [12] if everycongruence class C on a product X × Y in V is such that C = π ( C ) × π ( C ).This property is easily seen to be equivalent to the requirement that everycongruence Θ on any product X × Y in V , satisfies the implication:( x, y )Θ( x ′ , y ′ ) = ⇒ ( x ′ , y )Θ( x ′ , y ′ ) . Moreover, this may be viewed as the triangular lemma restricted to products:( x ′ , y ′ ) Eq( π ) ( x, y ) Θ ✈✈✈✈✈✈✈✈✈ Eq( π ) ( x ′ , y ) Θ Proposition 2.9.
Any pointed variety which has directly decomposable congru-ence classes is anticommutative.Proof.
We verify the conditions of Proposition 2.7 ( ii ). Suppose that f : X × X → Y is any morphism in V and that f ( x,
0) = f (0 , x ), then ( x, f )(0 , x )implies (0 , f )(0 , x ). (0 , x ) Eq( π ) ( x, Eq( f ) ✇✇✇✇✇✇✇✇ Eq( π ) (0 , Eq( f ) Remark 2.10.
Essentially the same argument, working categorically may beused to show that any pointed finitely complete category which satisfies thetriangular lemma is anticommutative. 10he notion of a majority category has been recently introduced and studied in[17, 20]. This notion is thought to be a categorical counterpart for varietieswhich admit a majority term, in a similar way the notion of Mal’tsev category[1] is a categorical counterpart of Mal’tsev varieties. A category C is a ma-jority category if for any ternary relation R between objects X, Y, Z we have:( x ′ , y, z ) ∈ S R and ( x, y ′ , z ) ∈ S R and ( x, y, z ′ ) ∈ S R implies ( x, y, z ) ∈ S R , forany morphisms x, x ′ : S → X and y, y ′ : S → Y and z, z ′ : S → Z in C . Accord-ing to Proposition 2.2 above, and Proposition 3.8 in [17], it follows that everypointed finitely complete majority category is anticommutative. However, forcompleteness we include a short proof here: Proposition 2.11.
Every pointed finitely complete majority category is anti-commutative.Proof.
We will show that ( i ) of Proposition 2.7 holds. Let f : X × X → Y beany morphism with f ◦ ι = y = f ◦ ι . Let R be the ternary relation defined bythe monomorphism r : R → X × Y × X where r is the equalizer of f ◦ ( π , π )and π . R r / / X × Y × X π / / f ◦ ( π ,π ) / / Y Then we have ( ι , y, ∈ X R and (0 , , ∈ X R and (0 , y, ι ) ∈ X R whichimplies (0 , y, ∈ X R by the majority property. Therefore, f ◦ ι = y = 0.The notion of an M -coextensive object [19] in a category C is an object-wisecoextensivity [10] condition relative to a class of morphisms M in C . Let C bea pointed category with binary products and coequalizers. When M is the classof regular epimorphisms in C , then an object X is M -coextensive if and only iffor any diagram X q (cid:15) (cid:15) X o o q (cid:15) (cid:15) / / (cid:15) (cid:15) X q (cid:15) (cid:15) Q Q / / o o Q where the top row is a product diagram and the vertical morphisms are regularepimorphisms, then the bottom row is a product diagram if and only if both thesquares above are pushouts. For example, given any algebra X in any variety,then X is M -coextensive (where M is the class of regular epimorphisms in thevariety) if and only it has the Fraser-Horn property [14]. We say that C isregularly-coextensive if every object in C is M -coextensive with M the class ofall regular epimorphisms in C . Proposition 2.12. If C is a pointed category with binary products and coequal-izers which is regularly coextensive, then C is anticommutative.Proof. We will show that C satisfies the conditions of Proposition 2.8. Let X be any object in C , and suppose that q : X × X → Q is a coequalizer of theproduct inclusions ι : X → X × X and ι : X → X × X . Then the pushout of q π and along π is formed simply by taking a coequalizer of π ◦ ι , π ◦ ι and π ◦ ι , π ◦ ι respectively. But these two pairs of coequalizers have terminalobjects for codomains, and hence Q being their product, is terminal. For an object X in a category, recall from the introduction that the fibre Pt C ( X )above X of the fibration of points π : Pt( C ) → C consists of triples ( A, p, s )where p : A → X is a split epimorphism in C and s is a splitting for p . Amorphism f : ( A, p, s ) → ( B, q, t ) in Pt C ( X ) is a morphism f : A → B in C suchthat q ◦ f = p and f ◦ s = t . The category Pt C ( X ) is always pointed, where thezero-object is ( X, X , X ), and if C is finitely complete, then so is Pt C ( X ). Definition 2.13.
A category C is locally anticommutative if for any object X in C , the category Pt X ( C ) is anticommutative. Proposition 2.14. If D is any finitely complete category which satisfies thetriangular lemma, and F : C → D is any conservative functor (,i.e., reflectsisomorphisms,) which preserves pullbacks and equalizers then C satisfies thetriangular lemma. Note that the assumptions on the functor F imply that it preserves monomor-phisms, and that if E is an equivalence relation in C , then F ( E ) – the relationobtained by applying F to the representative of E – is an equivalence relationin D . We provide a sketch of the proof below, which is a standard preserva-tion/reflection argument. Sketch.
Suppose that
R, S, T are equivalence relations on an object X in C suchthat R ∩ S T and let x, y, z : S → X be morphisms as in Definition 1.2, where xRySz and xT z . Then we are required to show that yT z , which is equivalentto showing that in the pullback diagram P p / / p (cid:15) (cid:15) T t (cid:15) (cid:15) S ( y,z ) / / X × Xp is an isomorphism. Applying F to the diagram above, we obtain a pull-back diagram in D . The assumptions on F easily imply that the canonicalmorphism F ( X × X ) → F ( X ) × F ( X ) is a monomorphism, which impliesthat ( F ( P ) , F ( p ) , F ( p )) form a pullback of F ( t ) along ( F ( y ) , F ( z )). Since F ( x ) F ( R ) F ( y ) F ( S ) F ( z ) and F ( x ) F ( T ) F ( z ) and since D satisfies the trian-gular lemma, ( F ( y ) , F ( z )) factors through T , which implies that F ( p ) is anisomorphism, so that p is an isomorphism since F reflects isomorphisms. Corollary 2.15. If C is a finitely complete category which satisfies the trian-gular lemma, then so does C ↓ X and X ↓ C for any object X . In particular, itfollows that Pt C ( X ) satisfies the triangular lemma if C does. roof. The proof follows from the fact that the codomain-assigning functor X ↓ C → C and the domain-assigning functors C ↓ X → C and Pt C ( X ) → C satisfythe conditions of Proposition 2.14. Remark 2.16.
The trapezoid lemma [11], which for varieties is equivalentto congruence distributivity, has also recently been studied in the categoricalsetting (see [28]). The same argument above, also applies to the trapezoidlemma, so that if a finitely complete category C satisfies the trapezoid lemma,then so does C ↓ X , X ↓ C and Pt C ( X ). Corollary 2.17.
Every finitely complete category C satisfying the triangularlemma is locally anticommutative.Proof. By Corollary 2.15 every category of points Pt C ( X ) satisfies the triangularlemma and is finitely complete, and the result follows by Remark 2.10. Example 2.2.
Every finitely complete majority category is locally anticommu-tative.
Proof. If C is any finitely complete majority category, then Pt C ( X ) is a pointedfinitely complete majority category (see Proposition 2.4 in [17]) for any object X in C , so that the result follows by Proposition 2.11.Much of what follows in the discussion below is just a slight adaption of theproof of Theorem 2.11 in [8] for local anticommutativity, and has been motivatedfrom the results of [30]. Our main aim in what follows is to provide sufficientcontext for Remark 2.23, and for this reason we keep the length of the discussionminimal so as to be self-contained.Let C be a finitely complete category. For any object X in C , we mayassociate the object ( X × X, π , ∆ X ) in Pt C ( X ), where π is the canonical pro-jection, and ∆ X is the diagonal morphism. Moreover, any equivalence relation E represented by a monomorphism e = ( e , e ) : E → X × X in C determinesan object ( E , e , d E ) in Pt C ( X ) where d E : X → E is the unique morphismwith e ◦ d E = ∆ X . Given any two equivalence relations R and S represented bythe monomorphisms r = ( r , r ) : R → X × X and s = ( s , s ) : S → X × X respectively, then the morphisms r and s can be viewed as monomorphisms( R , r , d R ) r −→ ( X × X, π , ∆ X ) and ( S , s , d S ) s −→ ( X × X, π , ∆ X ) in Pt C ( X ).In what follows, we will say that the equivalence relations R and S commute ,if the monomorphisms r and s commute in Pt C ( X ). An equivalence relation isthen said to be abelian if it commutes with itself. Consider the pullback below: R × X S p / / p (cid:15) (cid:15) S s (cid:15) (cid:15) R r / / X Set theoretically, the object R × X S may be considered as the subobject of X consisting of all triples ( x, y, z ) such that xRy and ySz . Any morphism p : R × X → X satisfying p ( x, x, y ) = y = p ( y, x, x ), i.e., any partial Mal’tsev operation p : R × X S → X , determines a cooperator for r and s in Pt C ( X ): let P be theproduct of ( R , r , d R ) and ( S , s , d S ) in Pt C ( X ), then the object of P , which isgiven by the pullback of r along s in C , may too be presented set-theoreticallyas the subobject of X × X consisting of all those pairs (( x, y ) , ( x, z )) such that xRy and xSz . Now, given a morphism p as above, the morphism φ : P → X × X defined by φ (( x, y ) , ( x, z )) = ( x, p ( y, x, z )) , is readily seen to be a cooperator for r and s in Pt C ( X ). Conversely, anycooperator for r and s determines a partial Mal’tsev operation p : R × X S → X by the formula above. Proposition 2.18.
Let C be a finitely complete locally anticommutative cate-gory. If R and S are any equivalence relations on the same object X in C whichcommute, then R ∩ S = ∆ X .Proof. This is immediate since R ∩ S is represented by the pullback of r along s in Pt C ( X ) (which must be the zero object since Pt C ( X ) is anticommutative),and the zero object in Pt C ( X ) is ( X, X , X ). Remark 2.19.
If the equivalence relations R and S admit a connector in thesense of [9], i.e., R and S are connected , then R and S commute. This isbecause a connector between R and S is a certain partial Mal’tsev operation p : R × X S → X satisfying some additional requirements. Definition 2.20.
An internal category C in a category C is a diagram C p / / p / / m / / C d / / d / / C s o o in C where the square C p / / p (cid:15) (cid:15) C d (cid:15) (cid:15) C d / / C is a pullback, and we have the following relations(i) d ◦ s = 1 C = d ◦ s (ii) m ◦ (1 C , s ◦ d ) = 1 C = m ◦ ( s ◦ d , C )(iii) d ◦ p = d ◦ m and d ◦ p = d ◦ m m ◦ ( p ◦ q , m ◦ q ) = m ◦ ( m ◦ q , p ◦ q ) where C q (cid:15) (cid:15) q / / C p (cid:15) (cid:15) C p / / C is a pullback, and ( p ◦ q , m ◦ q ) , ( m ◦ q , p ◦ q ) : C → C are themorphisms induced by the pullback ( C , p , p ). Definition 2.21.
An internal category C as in Definition 2.20 is called an internal groupoid if there exists an “inverse” morphism σ : C → C satisfyingthe following: d ◦ σ = d , d ◦ σ = d , and m ◦ (1 C , σ ) = s ◦ d , m ◦ ( σ, C ) = s ◦ d . Corollary 2.22.
Given any internal groupoid C as in Definition 2.20 and Defi-nition 2.21 in any finitely complete locally anticommutative category C , the mor-phism ( d , d ) : C → C × C is a monomorphism, i.e., any internal groupoid ina finitely complete locally anticommutative category is an equivalence relation.Proof. The groupoid structure determines a canonical connector between theequivalence relations Eq( d ) and Eq( d ) (see Example 1.4 in [9]), so that in par-ticular we have that Eq( d ) and Eq( d ) are connected, and hence Eq( d ) ∩ Eq( d ) = ∆ C by Proposition 2.18 and Remark 2.19. Then we have thatEq( d , d ) = Eq( d ) ∩ Eq( d ) = ∆ C and hence ( d , d ) is a monomorphism. Remark 2.23.
In the paper [31], M.C. Pedicchio showed that a congruencemodular variety V is congruence distributive if and only if every internal groupoidin V is an equivalence relation (see Theorem 3.2 in [31]). Note, this fact wasalready announced in [24]. In particular, it follows from Corollary 2.22 that acongruence modular variety is congruence distributive if and only if it is locallyanticommutative. By Theorem 3.4 (iii) below, we have that a variety which iscongruence modular and satisfies the triangular lemma on pullbacks is congru-ence distributive. In particular, this generalizes the equivalence of (a) and (d)of Theorem 1 in the paper [11]. Let V be a variety of universal algebras, and suppose that f : A → X and g : B → X are two homorphisms in V . The pullback A × X B of f along g isgiven by A × X B = { ( x, y ) | f ( x ) = g ( y ) } , p : ( x, y ) x and p : ( x, y ) y . As mentionedearlier, we note that the kernel congruence relation Eq( f ) of a homomorphism f in V is simply given by the pullback of f along itself. In the proofs of Theo-rem 3.1 and Theorem 3.4, we make use 2 × A × X B . In particular, this is represented asfollows: (cid:18) a a ′ b b ′ (cid:19) ∈ Θ ⇐⇒ ( a, b )Θ( a ′ , b ′ ) . Theorem 3.1.
For a pointed variety of universal algebras V , the following areequivalent.1. V is anticommutative.2. V admits unary terms u , . . . , u m and v , . . . , v m , as well as ( m + 2) -aryterms p , . . . , p n satisfying the equations: • p ( u ( x ) , . . . , u m ( x ) , x,
0) = x and p ( v ( x ) , . . . , v m ( x ) , , x ) = 0 . • p i +1 ( u ( x ) , . . . , u m ( x ) , x,
0) = p i ( u ( x ) , . . . , u m ( x ) , , x ) . • p i +1 ( v ( x ) , . . . , v m ( x ) , , x ) = p i ( v ( x ) , . . . , v m ( x ) , x, . • p n ( u ( x ) , . . . , u m ( x ) , , x ) = 0 and p n ( v ( x ) , . . . , v m ( x ) , x,
0) = 0 .Proof.
Let V be a pointed variety which satisfies ( ii ) of Proposition 2.7, and letΘ be the congruence on F V ( x ) × F V ( x ), generated by ( x, , x ). Then in thequotient of F V ( x ) × F V ( x ) by Θ, the element ( x,
0) will be identified with (0 , x, , { (cid:18) x x (cid:19) , (cid:18) xx (cid:19) } , under reflexivity, then under all operations, and then under transitivity. There-fore, there exists elements z , . . . , z n such that z = ( x,
0) and z n = (0 ,
0) and z i − Θ z i for all 0 < i . For each 0 < i we have that ( z i − , z i ) has the form: p i (cid:0) (cid:18) u ,i ( x ) u ,i ( x ) v ,i ( x ) v ,i ( x ) (cid:19) , . . . , (cid:18) u m i ,i ( x ) u m i ,i ( x ) v m i ,i ( x ) v m i ,i ( x ) (cid:19) , (cid:18) x x (cid:19) , (cid:18) xx (cid:19) (cid:1) , where the u ’s and the v ’s are unary terms. We may assume without loss ofgenerality that m i = m , and moreover, we may also assume that u k,i = u k as well as v k,i = v k for any i = 1 , . . . , m . Then writing out the equalitiescomponent-wise gives the equations in the statement of the theorem. Remark 3.2.
Recall that a pointed variety of universal algebras V is unital ifand only if it admits a J`onsson-Tarski operation (i.e., a binary term + satisfying x + 0 = x = 0 + x ). Therefore, by Proposition 2.4 a variety of universal algebrasis antilinear if and only if it admits a J`onsson-Tarski operation and the termsof Theorem 3.1. 16 .1 Characterization of locally anticommutative varieties Let C be any category with pullbacks, and let ( A, f, s ) be any object of Pt C ( X ).Consider the diagram below where the square is a pullback. X ( s,s ) ❊❊❊❊ " " ❊❊❊ s s $ $ Eq( f ) p / / p (cid:15) (cid:15) d ❊❊❊❊ " " ❊❊❊❊ A f (cid:15) (cid:15) A f / / X A binary product of (
A, f, s ) with itself in Pt C ( X ) is given by (Eq( f ) , d, ( s, s ))together with the morphisms p and p . If C has coequalizers, then so doesPt C ( X ), and moreover, they are computed as they are in C . Using these statedfacts, it is then possible to state a local version of Proposition 2.8, which is thecontent of the following proposition. Proposition 3.3.
Let C be a category with pullbacks and coequalizers. Then C is locally anticommutative if and only if for any split epimorphism f : A → X with splitting s : X → A and any coequalizer diagram A (1 ,s ◦ f ) / / ( s ◦ f, / / Eq( f ) q / / / / Q, we have q ( s ◦ f,
1) = q ( s ◦ f, s ◦ f ) = q (1 , s ◦ f ) . Recall from the introduction that we say a variety V satisfies the triangularlemma on pullbacks, if for any two morphisms f : A → X and g : B → X , andany congruence Θ on the pullback A × X B of f along g we have:( x, y )Θ( x ′ , y ′ ) = ⇒ ( x ′ , y )Θ( x ′ , y ′ ) , for any ( x, y ) , ( x ′ , y ) , ( x ′ , y ′ ) ∈ A × X B . ( x ′ , y ′ ) Eq( p ) Θ ( x, y ) Θ ✈✈✈✈✈✈✈✈✈ Eq( p ) ( x ′ , y ) Theorem 3.4.
For a variety V of universal algebras, the following are equiva-lent:(i) V is locally anticommutative.(ii) V admits binary terms b , . . . , b m and c , . . . , c m as well as ( m + 2) -aryterms p , . . . , p n such that: p ( b ( x, y ) , . . . , b m ( x, y ) , x, y ) = x , • p ( c ( x, y ) , . . . , c m ( x, y ) , y, x ) = y , • p i +1 ( b ( x, y ) , . . . , b m ( x, y ) , x, y ) = p i ( b ( x, y ) , . . . , b m ( x, y ) , y, x ) , • p i +1 ( c ( x, y ) , . . . , c m ( x, y ) , y, x ) = p i ( c ( x, y ) , . . . , c m ( x, y ) , x, y ) , • p n ( b ( x, y ) , . . . , b m ( x, y ) , y, x ) = x , • p n ( c ( x, y ) , . . . , c m ( x, y ) , x, y ) = x . • b i ( x, x ) = c i ( x, x ) (iii) V satisfies the triangular lemma on pullbacks.Proof. For ( i ) = ⇒ ( ii ), consider the free algebra F V ( x, y ) in V over the set { x, y } . Let f : F V ( x, y ) → F V ( z ) and s : F V ( z ) → F V ( x, y ) be the unique mor-phisms with f ( x ) = z = f ( y ) and s ( z ) = x , respectively. Consider the coequal-izer diagram F V ( x, y ) (1 ,s ◦ f ) / / ( s ◦ f, / / Eq( f ) q / / / / Q. Applying Proposition 3.3 to the diagram above, it follows that ( x, y )Eq( q )( x, x ).The coequalizer q is obtained by the quotient of Eq( f ) by the congruence gener-ated by the pairs ( s ◦ f, b ( x, y )) , (1 , s ◦ f )( b ( x, y )) where b is any binary termin the theory of V . It is straightforward to verify that (cid:0) ( s ◦ f, b ( x, y )) , (1 , s ◦ f )( b ( x, y )) (cid:1) = b (cid:0) (( x, x ) , ( x, x )) , (( x, y ) , ( y, x )) (cid:1) , and hence that Eq( q ) is the principle congruence generated by ( x, y )Eq( q )( y, x ).Therefore, there exists elements z , . . . , z n such that z = ( x, y ) and z n = ( x, x )and z i − Θ z i for 0 < i . Moreover, we have that ( z i − , z i ) is equal to p i (cid:0) (cid:18) b ( x, y ) b ( x, y ) c ( x, y ) c ( x, y ) (cid:19) , . . . , (cid:18) b m ( x, y ) b m ( x, y ) c m ( x, y ) c m ( x, y ) (cid:19) , (cid:18) x yy x (cid:19) , (cid:18) y xx y (cid:19) (cid:1) , for all 0 < i and where ( b i ( x, y ) , c i ( x, y )) ∈ Eq( f ). Then b i ( z, z ) = c i ( z, z ) andwriting out the equations determined by z i Θ z i +1 component-wise we get theequations in the statement of the theorem. For ( ii ) = ⇒ ( iii ), suppose that A × X B is the pullback of a morphism f : A → X along another morphisms g : B → X in V , and suppose that ( x, y ) , ( x, y ′ ) , ( x ′ , y ′ ) ∈ A × X B . Then f ( x ′ ) = f ( x ) = g ( y ) = g ( y ′ ), and hence f ( b i ( x, x ′ )) = b i ( f ( x ) , f ( x ′ )) = c i ( g ( y ) , g ( y ′ ) = g ( c i ( y, y ′ )) so that ( b i ( x, x ′ ) , c i ( y, y ′ )) ∈ A × X B for any i = 1 , . . . m . Define theelements ( z ,i , z ,i ) of Θ so that ( z ,i , z ,i ) is equal to p i (cid:0) (cid:18) b ( x, x ′ ) b ( x, x ′ ) c ( y, y ′ ) c ( y, y ′ ) (cid:19) , . . . , (cid:18) b m ( x, x ′ ) b m ( x, x ′ ) c m ( y, y ′ ) c m ( y, y ′ ) (cid:19) , (cid:18) x x ′ y y ′ (cid:19) , (cid:18) x ′ xy ′ y (cid:19) (cid:1) . Then the equations in the statement of the theorem ensure that z , = ( x, y )and z ,n = ( x ′ , y ), and that z ,i = z ,i +1 so that by the transitivity of Θ we have18 x, y )Θ( x ′ , y ). For ( iii ) = ⇒ ( i ), we have to show that for any split epimorphism p : A → X with splitting s : X → A , and any coequalizer X (1 ,s ◦ p ) / / ( s ◦ p, / / Eq( p ) q / / / / Q, that q ( s ◦ p,
1) = q ( s ◦ p, s ◦ p ) according to Proposition 3.3. But for any x ∈ X we have that ( s ◦ p ( x ) , x ) , ( x, s ◦ p ( x )) , ( s ◦ p ( x ) , s ◦ p ( x )) ∈ Eq( p ) and moreover wehave ( x, s ◦ p ( x ))Eq( q )( s ◦ p ( x ) , x ) which implies by ( iii ) that ( s ◦ p ( x ) , x )Eq( q )( s ◦ p ( x ) , s ◦ p ( x )) for any x , and the result follows. Recall from earlier that a variety V is said to have directly decomposable con-gruence classes (DDCC) [12] if every congruence class C on a product X × Y in V is such that C = π ( C ) × π ( C ). It was shown in [12] that (DDCC) isalso a Mal’tsev property, which turns out to be equivalent to the existence ofbinary terms b , . . . , b m and c , . . . , c m as well as ( m + 2)-ary terms p , . . . , p n such that: • p ( b ( x, y ) , . . . , b m ( x, y ) , x, y ) = x , • p ( c ( x, y ) , . . . , c m ( x, y ) , y, x ) = y , • p i +1 ( b ( x, y ) , . . . , b m ( x, y ) , x, y ) = p i ( b ( x, y ) , . . . , b m ( x, y ) , y, x ), • p i +1 ( c ( x, y ) , . . . , c m ( x, y ) , y, x ) = p i ( c ( x, y ) , . . . , c m ( x, y ) , x, y ), • p n ( b ( x, y ) , . . . , b m ( x, y ) , y, x ) = x , • p n ( c ( x, y ) , . . . , c m ( x, y ) , x, y ) = x .The above terms for (DDCC) differ from the terms in Theorem 3.4 by onlyone extra requirement, namely, that the b i ( z, z ) = c i ( z, z ). This observationimmediately gives us the following two corollaries of Theorem 3.4. Corollary 4.1.
Every locally anticommutative variety has (DDCC) , and anidempotent variety is locally anticommutative if and only if it has (DDCC) . Remark 4.2.
The above corollary implies that every locally anticommutativevariety of algebras has difunctional class relations in the sense of [21].Together with Remark 2.23 we also have:
Corollary 4.3.
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