higher geometry / derived geometry
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The notion of a topos that is equipped with a local algebra-object is a generalization of the notion of a locally ringed topos. The algebra object is then also called the structure sheaf.
For that reason in (Lurie) such pairs are called structured toposes. But since the notion of locally ringed topos is a special case, maybe a more systematic and descriptive term is locally algebra-ed topos. Elsewhere this is called a locally -modelled topos, where the given algebraic theory.
Let be the syntactic category of an essentially algebraic theory , hence any category with finite limits. Let be a subcanonical coverage on . Notice that this makes be a standard site and every standard site will do.
Then the sheaf topos is the classifying topos for the geometric theory of -local algebras.
For any topos, a local -algebra object in is a geometric morphism
By the discussion at classifying topos this is equivalently a functor
such that
it preserves finite limits (and hence produces a -algebra in );
it sends -coverings to epimorphisms; which makes it a local -algebra.
The pair is called a locally -algebra-ed topos.
All of the following notions are special cases of locally algebra-ed toposes:
The (∞,1)-category theory-version is that of
Julian Cole, The Bicategory of Topoi, and Spectra , ms. (pdf)
Last revised on October 11, 2014 at 16:15:30. See the history of this page for a list of all contributions to it.