On Exponentiable Morphisms in Classical Algebra

We study exponentiability of homomorphisms in varieties of universal algebras close to classical ones. After describing an “almost folklore” general result, we present a purely algebraic proof of “étale implies exponentiable”, alternative to the topologically motivated proof given in one of our previous papers, in a different context. We prove that only isomorphisms are exponentiable homomorphisms in ideal determined varieties and extend this to ideal determined categories. Finally, we give a complete characterization of exponential homomorphisms of semimodules over semirings. Introduction Recall that a morphism f : A → B in a category C is said to be exponentiable if the pullback functor f∗ : (C ↓ B) → (C ↓ A) has a right adjoint. This paper is one of many that attempt to characterize such morphisms in various concrete situations. The categories we are interested in are varieties of universal algebras, which, being close to those of classical algebra, have few exponentiable morphisms. More specifically, the four sections of this paper are devoted to the following four questions respectively: Question 0.1. What can we say about exponentiability in a variety of universal algebras in general? Since P. T. Johnstone [18] says, after giving an object-wise characterization of cartesian closed varieties, “...the argument of the above proof may be used to characterize the exponentiable objects of T -Alg...” one might expect that we aim at a syntactical characterization of exponentiable homomorphisms of algebras. We do not go that far, but only make preliminary remarks, the most important of which is that the exponentiability of f reduces to preservation of finite coproducts by the functor f∗ – in fact even just to preservation of finite coproducts of objects in (C ↓ B) with free domains. This “almost folklore” result is a natural counterpart of Proposition 3.1 in [18], to whose proof the citation above refers. 1991 Mathematics Subject Classification. 18C15, 18C20, 08A62, 16Y60.