Anticommutativity and n-schemes

Abstract The purpose of this paper is two-fold. A first and more concrete aim is to give new characterizations of equivalence distributive Goursat categories (which extend 3-permutable varieties) through variations of the little Pappian Theorem involving reflexive and positive relations. A second and more abstract aim is to show that every finitely complete category E satisfying the n-scheme is locally anticommutative.


Introduction and Preliminaries
In this section we recall some basic definitions and results from the literature, needed throughout the article.

n-schemes
For a sublattice L of an equivalence lattice EqA, Gumm's Shifting Lemma [11] is stated as follows. Given congruences R, S and T on the same algebra X in V such that R∧S T , whenever x, y, z, t are elements in X with (x, y) ∈ R ∧ T , (x, t) ∈ S, (y, z) ∈ S and (t, z) ∈ R, it then follows that (t, z) ∈ T . We display this condition as x S R t S 2 , (x , u), (u, z), (y , x), (x, z ) ∈ R and (z, z ) ∈ T, then (x , y ) ∈ T : Similarly, on identifying S 2 with T and u with z we obtain A sublattice L of EqA satisfies the scheme-1 if given congruences R, S and T on the same algebra X in L such that R ∧ S T , one has

Anticommutative categories
Our categories will always be regular, in the sense of Barr [2]; we recall that a category is regular if it has finite limits, each arrow factors as a regular epi followed by a mono, and regular epis are pull-back stable. (It turns out that in a regular category the kernel pair of an arrow always has a coequalizer, given by the regular epi part of the factorization of the arrow) In a regular category, it is possible to compose relations. If (R, r 1 , r 2 ) is a relation from X to Y and (S, s 1 , s 2 ) a relation from Y to Z, their composite SR is a relation from X to Z obtained as the regular image of the arrow where (R × Y S, π 1 , π 2 ) is the pullback of r 2 along s 1 . The composition of relations is then associative, thanks to the fact that regular epimorphisms are assumed to be pullback 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. A point in a category E is a split epimorphism p : A → X together with a fixed splitting s : X → A, usually depicted as Let E be an arbitrary category. The category P t E (X) [3] of points of E over X is the category of pointed objects of the comma category E ↓ X, that is, Explicitly, objects of this category are triples (A, p, s) where A is an object of E and p : A → X and s : and if E is finitely complete, then so is P t E (X). Recall that two morphisms f : A → C and g : B → C in a pointed category E with binary products are said to commute [15] if there exists a morphism ρ : This brings us to the main definition of this paper: A pointed category E with binary products is a called anticommutative if every pair of commuting morphisms are disjoint.

Majority Categories and Goursat Categories
For a regular category E the property of being a majority category can be equivalently defined as follows (see [16]): for any reflexive relations R, S and T on the same object holds. We then observe that any regular majority category satisfies the 3-scheme and, consequently, also the 2-scheme and Shifting Lemma): Lemma 2.1. The n-scheme holds true in any regular majority category E.
Proof. Given equivalence relations R, S and T on the same object such that R ∧ S T , for n even. Here (S, T ) n denotes the composite ST ST · · · of S and T , n times.
Corollary 2.2. Let E be a regular majority category.
(1) The little Pappian Theorem holds true in E.
A variety V of universal algebras is called 3-permutable when the strictly weaker equality RSR = SRS holds. Such varieties are characterized by the existence of two quaternary operations p and q satisfying the identities p(x, y, y, z) = x, p(u, u, v, v) = q(u, u, v, v), q(x, y, y, z) = z (see [10]). The notions of 3-permutability can be extended from varieties to regular categories by replacing congruences with (internal) equivalence relations, allowing one to explore some interesting new (non-varietal) examples. Regular categories that are 3-permutable are usually called Goursat categories. As examples of Goursat categories we have :compact groups, topological groups, torsion-free abelian groups, reduced commutative rings. It is well-known that any 3-permutable variety is congruence modular, thus the Shifting Lemma and 3-scheme hold. This result also extends to the regular categorical context. (i) E is a Goursat category; (ii) ∀R, S ∈ Equiv(X), RSR = SRS ∈ Equiv(X), for any X; (iii) every relation P → X × Y in E, P P • P P • = P P • ; (iv) every reflexive relation F in E, F • F = F F • ∈ Equiv(X); (v) every reflexive and positive relation in E is an equivalence relation.
Let us begin with the following observation: (1) The Little Pappian Theorem holds true in E when S i is a reflexive relation and R and T are equivalence relations.
(2) The scheme-1 holds true in E when S is a reflexive relation and R and T are equivalence relations.
Proof. The proof of this result is based on that of Proposition 5.3 in [12] which claims that a Goursat category satisfies the Shifting Lemma, 2-scheme and 3-scheme when S is a reflexive relation and R and T are equivalence relations. We prove (1). Let R and T be equivalence relations and let S i be a reflexive relation on an object X such that R ∧ S i T . Suppose that x, y, z, u, x , y , z are elements in X related as in (1.1). We are going to show that (x , y ) ∈ T . We apply 2-scheme to z We now apply the Shifting Lemma to We now apply the Shifting Lemma to It follows that,(x , u), (u, z) ∈ T and (z, z ), (z , x) ∈ T , (x, y ) ∈ T . We conclude that x T y (T is transitive), as desired.
We are now ready to prove the main result in this section: Theorem 2.5. Let E be a regular category. The following conditions are equivalent: (1) E is an equivalence distributive Goursat category; (2) the Little Pappian Theorem holds true in E when S i is a reflexive relation and R and T are reflexive and positive relations;  Theorem 2.3 (iv)). Suppose that (x, y) ∈ EE • . Then, for some z in X, one has that (z, x) ∈ E and (z, y) ∈ E. Consider the reflexive and positive relations R = EE • and T = E • E, and the reflexive relation E on X. Then we have:

Locally Anticommutative Categories
The fibration of points π : P t(E) → E classifies many central notions in categorical algebra, such as, Mal'tsev categories:a finitely complete category E is Mal'tsev if and only if every fibre P t E (X) of the fibration of points is unital, strongly unital or subtractives [3].
Definition 3.1. [15] A category E is locally anticommutative if for any object X in E, the category P t E (X) is anticommutative.
Proposition 3.2. If D is any finitely complete category which satisfies the n-scheme and U : E → D is any conservative functor (i.e., reflects isomorphisms) which preserves pullbacks and equalizers, then E satisfies the n-scheme.
Note that the assumptions on the functor U imply that it preserves monomorphisms, and that if R is an equivalence relation in E, then U (R) the relation obtained by applying U to the representative of R is an equivalence relation in D.
Proof. Let R, S, T are equivalence relations on an object X in E such that R ∧ S T and for x, y, z 1 , · · · , z n are related such that x, y ∈ R, x, z 1 ∈ S, z 1 , z 2 ∈ T, z 2 , z 3 ∈ S, · · · , z n−1 , y ∈ S for n odd and z n−1 , y ∈ T for n even. Then we are required to show that xT y, which is equivalent to showing that in the pullback diagram / / X × X p 2 is an isomorphism. Applying U to the diagram above, we obtain a pullback diagram in D. The assumptions on U easily imply that the canonical morphism U (X × X) → U (X) × U (X) is a monomorphism, which implies that (U (P ), U (p 1 ), U (p 2 )) form a pullback of U (t) along (U (x), U (y)). Since for n odd and U (z n−1 ), U (y) ∈ U (T ) for n even. Since D satisfies the n-scheme (U (x), U (y)) factors through T , which implies that U (p 2 ) is an isomorphism, so that p 2 is an isomorphism since U reflects isomorphisms. (i) If E is a finitely complete category which satisfies the n-scheme, then so does E ↓ X and X ↓ E for any object X. In particular, it follows that P t E (X) satisfies the n-scheme if E does.
(ii) Every finitely complete category E satisfying the n-scheme is locally anticommutative.
Proof. The proof follows from the fact that the codomain-assigning functor X ↓ E → E and the domain-assigning functors E ↓ X → E and P t C (X) → E satisfy the conditions of Proposition 3.2.
Proposition 3.4. If D is any finitely complete category which satisfies the the scheme-1 and U : E → D is any conservative functor (i.e., reflects isomorphisms) which preserves pullbacks and equalizers, then E satisfies the scheme-1.
Proof. Let R, S, T are equivalence relations on an object X in E such that R ∧ S T and for x, y, z, x , u, y , z are related as follows we show that x T y . We apply Proposition 3.2 (3-scheme) to x R S T z S z T x Next, We apply Proposition 3.2 (2-scheme) to y R S u T y T It follows that,(x , z ) ∈ T, (z , y) ∈ T and (y, y ) ∈ T , we conclude that x T y (T is transitive), as desired. (i) If E is a finitely complete category which satisfies the scheme-1, then so does E ↓ X and X ↓ E for any object X. In particular, it follows that P t E (X) satisfies the the scheme-1 if E does.
(ii) Every finitely complete category E satisfying the the scheme-1 is locally anticommutative.