nLab
deformation context

Contents

Context

Small objects

Higher geometry

Contents

Definition

Definition

A deformation context is an (∞,1)-category 𝒴\mathcal{Y} such that

  1. it is a presentable (∞,1)-category;

  2. it contains a terminal object

together with a set of objects {E αStab(𝒴)}\{E_\alpha \in Stab(\mathcal{Y})\} in the stabilization of 𝒴\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.

Remark

Definition is meant to be read as follows:

First, we think of 𝒴\mathcal{Y} as an opposite (∞,1)-category of pointed spaces in some higher geometry. The point is the initial object in 𝒴 op\mathcal{Y}^{op} which is the terminal object in 𝒴\mathcal{Y}.

Then we think of the formal duals of the objects {E α} α\{E_\alpha\}_\alpha as a set of generating infinitesimally thickened points.

The following construction generates the “jets” induced by the generating infinitesimally thickened points.

Definition

Given a deformation context (𝒴,{E α} α)(\mathcal{Y}, \{E_\alpha\}_\alpha), we say

  • a morphism in 𝒴\mathcal{Y} is an elementary morphism if it is the homotopy fiber to a map into Ω nE α\Omega^{\infty -n}E_\alpha for some nn \in \mathbb{Z} and some α\alpha;

  • a morphism is a small morphism if it is the composite of finitely many elementary morphisms.

We write

𝒴 inf𝒴 \mathcal{Y}^{inf} \hookrightarrow \mathcal{Y}

for the full sub-(∞,1)-category on those objects AA for which the essentially unique map A*A \to * is small.

(Lurie, def. 1.1.8)

References

Created on February 7, 2013 at 17:40:12. See the history of this page for a list of all contributions to it.