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 be an essentially algebraic theory and write for its syntactic category: the category of finitely presented -algebras
Let be a coverage on .
For a topos, a -local -algebra in is a functor
that
preserves finite limit;
sends -coverings in to epimorphisms in .
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
Created on April 27, 2011 at 14:37:50. See the history of this page for a list of all contributions to it.