symmetric monoidal (∞,1)-category of spectra
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 be a coverage on .
A topos equipped with a local algebra object is a locally algebra-ed topos.
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