nLab
sheaf of L-∞ algebras

Idea

A sheaf of $L_\infty$-algebras is a suitable sheaf/stack/∞-stack of L-∞ algebras/dg-Lie algebras.

These structures appear in the deformation theory of the given base space/site: where a single L-∞ algebra is equivalently a formal moduli problem (hence “over the point), so a sheaf of these over some base $X$ may be thought of as an $X$-parameterized formal moduli problem (eg. Hinich 03).

Related concepts

• descent for L-∞ algebras

• L-∞ algebroid

• homotopy theory of L-∞ algebras

• sheaf of spectra

References

Discussion in the context of deformation theory/parameterized formal moduli problems is in

• Vladimir Hinich, Deformations of sheaves of algebras (arXiv:math/0310116)

Quasi-coherent sheaves of dg-Lie algebras appear in

• Amnon Yekutieli, appendix A of Twisted Deformation Quantization of Algebraic Varieties (arXiv:0905.0488)

• I. Ciocan-Fontanine, Mikhail Kapranov, Derived Quot schemes (arXiv:math/9905174)

Discussion in physics, of sheaves of $L_\infty$-algebras of physical fields in prequantum field theory includes

• Anton Zeitlin, Conformal Field Theory and Algebraic Structure of Gauge Theory (arXiv:0812.1840)