Algebraic theories and commutativity in a sheaf topos

Abstract

For any site of definition $\mathcal{C}$ of a Grothendieck topos $\mathcal{E}$, we define a notion of a $\mathcal{C}$-ary Lawvere theory $\tau :\mathcal{C}\to \mathcal{T}$ whose category of models is a stack over $\mathcal{E}$. Our definitions coincide with Lawvere's finitary theories when $\mathcal{C}={\aleph }_0$ and $\mathcal{E}=\mathrm{Set}$. We construct a fibered category ${\mathrm{Mod}}^{\mathcal{T}}$ 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 $F⊣U:{\mathrm{Mod}}^{\mathcal{T}}\rightleftarrows \mathcal{E}$. If $\tau$ 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.

paper has been substantially reorganized