Finitary quasi-varieties

A pair (C,U) consisting of a category C with coequalizers and a functor U: C → Set is a weak quasi-variety if U has a left adjoint and U preserves and reflects regular epis. It is known that every weak quasi-variety is equivalent to a concrete quasi-variety, i.e. a category of Σ-algebras which has all free algebras and which is closed with respect to products and subalgebras. It is also known that if U preserves monic direct limits, C is equivalent to a concrete quasi-variety of Σ-algebras in which Σ contains no function symbols of infinite rank; and if U preserves all direct limits, C is equivalent to a concrete quasi-variety of Σ-algebras definable by a set of implications of the form (t1 = s1∧⋯∧tm = sm) ⇒ tm+1 = sm+1 where ti and si are Σ-terms and m is a nonnegative integer. This paper concerns several definitions of 'finiteness' in a category theoretic setting and some theorems on weak quasi-varieties. Two main theorems characterize those weak quasi-varieties (C, U) such that U preserves all direct limits. © 1982.