symmetric monoidal (∞,1)-category of spectra
A local algebra over an algebraic theory is to an algebra over an algebraic theory as a local ring is to a ring:
a local algebra in a sheaf topos is an algebra object / sheaf of algebras, which is determined by its local restrictions, for a sense of local determined both by the Grothendieck topology of any site of definition of the topos, as well as by a coverage on the category of finitely presented algebras.
Let $\mathbb{T}$ be an essentially algebraic theory and write $\mathcal{C}_{\mathbb{T}}$ for its syntactic category: the category of finitely presented $\mathbb{T}$-algebras
Let $J$ be a coverage on $\mathcal{C}_{\mathbb{T}}$.
For $\mathcal{E}$ a topos, a $J$-local $\mathbb{T}$-algebra in $\mathcal{E}$ is a functor
that
preserves finite limit;
sends $J$-coverings in $\mathcal{C}_{\mathbb{T}}$ to epimorphisms in $\mathcal{E}$.
A topos equipped with a local algebra object is a locally algebra-ed topos.
A theory of local algebras is a geometric theory and every geometric theory is the theory of some local algebras.
For the moment see classifying topos for details.
A local ring is a local algebra for the theory of rings.
A topos equipped with a local ring is a locally ringed topos.
The (∞,1)-category theory-analog of a theory of local algebras is (except for the additional choice of “admissible morphisms”) a