2018/03/26 by Haberman, Boaz
#Analysis of PDEs (math.AP) #Category Theory (math.CT) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1803.09378
For any site of definition \mathcal C of a Grothendieck topos \mathcal E, we define a notion of a \mathcal C-ary Lawvere theory τ: \mathscr C → \mathscr T whose category of models is a stack over \mathcal E. Our definitions coincide with Lawvere's finitary theories when \mathcal C=ℵ0 and \mathcal E = \operatorname\mathbf Set. We construct a fibered category \operatorname\mathbf Mod\mathscr T of models as a stack over \mathcal E and prove that it is \mathcal E-complete and \mathcal E-cocomplete. We show that there is a free-forget adjunction F \dashv U: \operatorname\mathbf Mod\mathscr T \rightleftarrows \mathscr E. If τ 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 \mathcal E-categories. Our results give a general recipe for constructing a monoidal \mathcal E-cosmos in which one can do enriched \mathcal E-category theory. As an application, we describe a convenient category of linear spaces generated by the theory of Lebesgue integration.