  For any site of definition $\mathcal C$ of a Grothendieck topos $\mathcal E$,
we define a notion of a $\mathcal C$-ary Lawvere theory $\tau: \mathscr C \to
\mathscr T$ whose category of models is a stack over $\mathcal E$. Our
definitions coincide with Lawvere's finitary theories when $\mathcal
C=\aleph_0$ and $\mathcal E = \operatorname{\mathbf {Set}}$. 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 $\tau$ 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.
