Algebra in a Grothendieck topos: Injectivity in quasi-equational classes

AbstractThis paper gives a study of injectivity and some related notions in quasi-equational classes of algebras in an arbitrary Grothendieck topos E. The main purpose is to describe the relationship between the class mod Σ of models of a set Σ of quasi-equations in the category of sets Ens and the corresponding class mod(Σ, E) of models of Σ in E with respect to residual smallness, boundedness of essential extensions and injectivity. The basic nature of our results is that, for any given Σ, whatever holds in Ens, concerning these notions, also holds in E. In particular, this substantially improves the earlier results of Howlett [5] regarding the existence of enough injectives in mod(Σ, E)