Algebraic Theories and Commutativity in a Sheaf Topos

Abstract

For any site of definition $$\mathcal {C}$$ C of a Grothendieck topos $$\mathcal {E}$$ E, we define a notion of a $$\mathcal {C}$$ C-ary Lawvere theory $$\tau : \mathscr {C} \rightarrow \mathscr {T}$$ τ:C→T whose category of models is a stack over $$\mathcal {E}$$ E. Our definitions coincide with Lawvere’s finitary theories when $$\mathcal {C}=\aleph _0$$ C=ℵ0 and $$\mathcal {E} = {{\,\mathrm{\mathbf {Set}}\,}}$$ E=Set. We construct a fibered category $${{\,\mathrm{\mathbf {Mod}}\,}}^{\mathscr {T}}$$ ModT of models as a stack over $$\mathcal {E}$$ E and prove that it is $$\mathcal {E}$$ E-complete and $$\mathcal {E}$$ E-cocomplete. We show that there is a free-forget adjunction $$F \dashv U: {{\,\mathrm{\mathbf {Mod}}\,}}^{\mathscr {T}} \leftrightarrows \mathscr {E}$$ F⊣U:ModT⇆E. If $$\tau $$ 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 $$\mathcal {E}$$ E-categories. Our results give a general recipe for constructing a monoidal $$\mathcal {E}$$ E-cosmos in which one can do enriched $$\mathcal {E}$$ E-category theory. As an application, we describe a convenient category of linear spaces generated by the theory of Lebesgue integration.