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created: 2023-03-26T11:58
updated: 2023-04-14T16:45
aliases: ["deformations are controlled by a DGLA"]
---

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- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [formal moduli problem](Maurer-Cartan.md)
	- [Tangent complex](cotangent%20complex.md#Tangent%20complex)

---

# deformations are controlled by a DGLA

![](2023-04-02-1.png)

![](2023-04-14-8.png)

The functor $\Phi^{-1}: \operatorname{Lie}_k^{\mathrm{dg}} \rightarrow$ Moduli $\subseteq \operatorname{Fun}\left(\mathrm{CAlg}_{\mathrm{sm}}, \mathcal{S}\right)$ is constructed by Hinich. Roughly speaking, if $\mathfrak{g}$ is a [DGLA](L%20infty%20algebra.md) and $A$ is a small $E_{\infty}$-algebra over $k$, then $\Phi^{-1}(\mathfrak{g})(A)$ is the space of solutions to the Maurer-Cartan equation $d x=[x, x]$ in the [DGLA](L%20infty%20algebra.md) $\mathfrak{g} \otimes_k \mathfrak{m}_A$.