higher geometry / derived geometry
Ingredients
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geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
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derived smooth geometry
Theorems
A deformation context is an (∞,1)-category $\mathcal{Y}$ such that
it is a presentable (∞,1)-category;
it contains a terminal object
together with a set of objects $\{E_\alpha \in Stab(\mathcal{Y})\}$ in the stabilization of $\mathcal{Y}$.
This is (Lurie, def. 1.1.3) together with the assumption of a terminal object stated (and later implicialy used) on p.9.
Definition is meant to be read as follows:
First, we think of $\mathcal{Y}$ as an opposite (∞,1)-category of pointed spaces in some higher geometry. The point is the initial object in $\mathcal{Y}^{op}$ which is the terminal object in $\mathcal{Y}$.
Then we think of the formal duals of the objects $\{E_\alpha\}_\alpha$ as a set of generating infinitesimally thickened points.
The following construction generates the “jets” induced by the generating infinitesimally thickened points.
Given a deformation context $(\mathcal{Y}, \{E_\alpha\}_\alpha)$, we say
a morphism in $\mathcal{Y}$ is an elementary morphism if it is the homotopy fiber to a map into $\Omega^{\infty -n}E_\alpha$ for some $n \in \mathbb{Z}$ and some $\alpha$;
a morphism is a small morphism if it is the composite of finitely many elementary morphisms.
We write
for the full sub-(∞,1)-category on those objects $A$ for which the essentially unique map $A \to *$ is small.
Created on February 7, 2013 at 17:40:12. See the history of this page for a list of all contributions to it.