Algebraic Theories and Commutativity in a Sheaf Topos

摘要

For any site of definition C of a Grothendieck topos E, we define a notion of a C-ary Lawvere theory t : C. T whose category of models is a stack over E. Our definitions coincide with Lawvere's finitary theories when C =. 0 and E = Set. We construct a fibered category ModT of models as a stack over E and prove that it is E-complete and E-cocomplete. We showthat there is a free-forget adjunction F similar to U : ModT similar to E. If t is a commutative theory in a certain sense, then we obtain a "locally monoidal closed" structure on the category of models, which enhances the free-forget adjunction to an adjunction of symmetric monoidal E-categories. Our results give a general recipe for constructing a monoidal E-cosmos inwhich one can do enriched E-category theory. As an application, we describe a convenient category of linear spaces generated by the theory of Lebesgue integration.