Algebra in a Topos of Sheaves

Abstract

<p>In this thesis, we undertake the study of some classical set-based algebraic concepts in a topos-theoretic setting. Actually, the topoi we are particularly interested in are the Grothendieck topoi.</p> <p>The main topics from Universal Algebra considered here are injectivity, equational compactness and tensor products.</p> <p>After proving some general results about the above notions, we show that, for any set of ℋ quasi-equations and an arbitrary Grothendieck topos E, ℳod(ℋ,E) has enough injectives iff ℳod ℋ has. Also that, for a noetherian Locale ℒ, pure homomorphisms, equational compactness and the existence of equationally compact hulls are characterized here the same way as in Ens. Finally, we consider the notion of bimorphisms for algebras in topos and prove, among other things, the counterpart of a result for algebras in Ens that tensor products and Universal bimorphisms are equivalent for suitable categories of algebras.</p>