nLab deformation context

Definition

Definition

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

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

2. 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.

Remark

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.

Definition

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

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

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

(Lurie, def. 1.1.8)

Related concepts

• deformation theory

• formal moduli problem

• Lie differentiation

• model structure for L-infinity algebras

References

• Jacob Lurie, Formal moduli problems