The category 𝒮hjIMSet of sheaves in MSet

Abstract

Since topoi were introduced, there have been efforts putting mathematics into the context of topoi. Amongst known topoi, the topoi of sheaves or presheaves over a small category are of special interest. We have here as the base topos that of sheaves over a monoid [Formula: see text] as a one object category. By means of closure operators we then obtain categories of sheaves related to the right ideals of [Formula: see text]. These categories have already been studied but we give these categories a more thorough treatment and reveal some additional properties. Namely, for a weak topology determined by a right ideal [Formula: see text] of [Formula: see text], we show that the category of sheaves associated to this topology is a subtopos of [Formula: see text] (the presheaves over [Formula: see text]) and determine the Lawvere–Tierney topology yielding the same subtopos, which is the Lawvere–Tierney topology associated to the idempotent hull of the (not necessarily idempotent) closure operator associated to [Formula: see text]. We will then find conditions under which the subcategory of separated objects turns out to be a topos, and in the last section, we find conditions under which the category of sheaves becomes a De Morgan topos.