Characterization of protomodular varieties of universal algebras.pdf

Theory and Applications of Categories, Vol. 11, No. 6, 2003, pp. 143–147. CHARACTERIZATION OF PROTOMODULAR VARIETIES OF UNIVERSAL ALGEBRAS DOMINIQUE BOURN AND GEORGE JANELIDZE ABSTRACT. Protomodular categories were introduced by the first author more than ten years ago. We show that a variety V of universal algebras is protomodular if and only if it has 0-ary terms e1, . . . , en, binary terms t1, . . . , tn, and (n+1)-ary term t satisfying the identities t(x, t1(x, y), . . . , tn(x, y)) = y and ti(x, x) = ei for each i = 1, . . . , n. 1. Introduction Protomodular categories were first introduced in [2]; their role in algebra, and various further developments are also described in [3]-[6]. Recall that if C is a category and B is any object in it, then Pt(B) denotes the category of points in the slice category C/B, i.e. the category whose objects are the triples (A,α, β) in which α : A → B and β : B → A are morphisms in C with α.β = 1B, and whose morphisms are the commutative triangles between such points over B. When C has finite limits, any morphism p : E → B in C determines a pullback functor p?: p? : Pt(B) → Pt(E) (1.1) Then the category C is said protomodular when, for every morphism p, the functor p? is conservative, i.e. reflects isomorphisms. Whenever C has an initial object 0, it obviously suffices to require the functor (1.1) to reflect isomorphisms just for the initial object E = 0. And then, if C is pointed (and so 0 = 1 in C), that requirement transforms into the so-called Split Short Five Lemma. In particular, the category of groups is protomodular [2]. A simple means of producing new examples comes from the fact that every category that admits a pullback preserving conservative functor from it into a protomodular category, is protomodular itself. There- fore any variety of groups with additional algebraic structure (like rings and modules or algebras over rings, etc.) also is protomodular. Thanks to the Yoneda embedding, the same is true for the intern