Products in categories of fractions and universal inversion of homomorphisms

Abstract

Main result. If finite direct products exist in a category k and the class of morphisms Σ is such that the category of fractions k[Σ -1] exists, where σ ∈ Σ implies that σ × 1 X ∈ Σ and 1 X × σ ∈ Σ for any objects X, then finite direct products also exist in k[Σ -1] and the canonical functor k → k[Σ -1] preserves these products. Using this theorem analogues of the theory of matrix localization of rings are constructed for arbitrary varieties of universal algebras and for preadditive categories.