symmetric monoidal (∞,1)-category of spectra
equality (definitional, propositional, computational, judgemental, extensional, intensional, decidable)
isomorphism, weak equivalence, homotopy equivalence, weak homotopy equivalence, equivalence in an (∞,1)-category
Examples.
For $(\mathfrak{g}, [-,-],\partial)$ a dg-Lie algebra (the differential of degree -1), a Maurer-Cartan element in $\mathfrak{g}$ is
an element $a \in \mathfrak{g}_1$ of degree -1
such that the Maurer-Cartan equation holds
This is a special case of MC elements for L-∞ algebras, which we discuss next.
For $\mathfrak{g}$ an L-∞ algebra with brackets $\{[-,\cdots,-]_k\}_k$, a Maurer-Cartan element is an element $a \in \mathfrak{g}$ such that
For $\mathfrak{g}$ an L-∞ algebra and $A$ a dg-algebra, also the tensor product $A \otimes \mathfrak{g}$ naturally inherits the structure of an L-∞ algebra.
Let $\mathfrak{g}$ be of finite type and write $CE(\mathfrak{g})$ for the Chevalley-Eilenberg algebra of $\mathfrak{g}$. Then MC-elements in $A \otimes \mathfrak{g}$ correspond bijectively to dg-algebra homomorphisms $A \leftarrow CE(\mathfrak{g})$:
A reference for this is for instance around def. 3.1 in (Hain).
We unwind in steps how this comes about.
The space of graded algebra homomorphisms $A \leftarrow CE(\mathfrak{g})$ is a subspace of the space of linear maps of graded vector spaces, and since $\mathrm{CE}(\mathfrak{g})$ is freely generated as a graded algebra and is of finite type by assumption, this is isomorphic to the space of grading preserving homomorphisms
of linear grading-preserving maps from the graded vector space $\mathfrak{g}^*$ of dual generators to $A$. By the usual relation in $\mathrm{Vect}[\mathbb{Z}]$ for $\mathfrak{g}$ of finite type, this is isomorphic to the space of elements of total degree degree 1 in elements of $A$ tensored with $\mathfrak{g}$:
The dg-algebra homomorphism form the subspace of this space
on elements that respect the differential. Under the above equivalence this are elements $a$ in $(A \otimes \mathfrak{g})_1$ satisfying a certain condition. By inspection one finds that this condition is precisely the MC equation
For instance if $A = \Omega^\bullet(X)$ is the de Rham algebra of a smooth manifold $X$, then $MC(A \otimes \mathfrak{g})$ is the space of flat L-∞ algebra valued differential forms on $X$. See there for more details.
For $\mathfrak{g}$ a Lie algebra, $X$ a smooth manifold, there is a canonical dg-Lie algebra structure on $\Omega^\bullet(X) \otimes \mathfrak{g}$.
A Maurer-Cartan element is then precisely a Lie algebra valued 1-form $A$ whose curvature 2-form vanishes
Maurer–Cartan equation is a name for very many related equations in geometry, algebra, deformation theory, category theory and quantization theory?. Such equations express for example certain conditions in theory of isometric embedding of submanifolds into a euclidean space (‘structure equations’, with relations to the Lie groups $O(n)$), invariance of invariant differential forms (Maurer-Cartan forms) on Lie groups, flatness of connections on principal or associated fibre bundles, the solutions in some contexts parametrize infinitesimal deformations, or define twisting cochains. In the context of BV-quantization, a Maurer–Cartan equation has the role of classical master equation.
A Maurer–Cartan equation for $A_\infty$-algebras is usually referred to as a generalized Maurer–Cartan equation as it has more summands than the one for dg-algebras. In some contexts like $A_\infty$-categories, some authors prefer the geometric terminology ‘homological vector field’ as a datum on a formal geometric space which satisfies a Maurer–Cartan equation. Solutions to Maurer-Cartan equation for a dg- or $A_\infty$ algebra are called Maurer-Cartan elements.
Sophus Lie considered groups of transformations first and discovered Lie algebras only later (letter to Mayer, 1874). He has shown that infinitesimally one can solve the Maurer–Cartan equations for a given set of structure constants of a finite-dimensional Lie algebra. This means that one can construct a neighborhood with either the invariant differential form, or dually the invariant vector fields whose commutator corresponds to the commutator of the Lie algebra. This amounts to integrating the Lie algebra to a local Lie group. Only much later, Elie Cartan succeeded in proving the global version of integration, that is the Cartan–Lie theorem. J-P. Serre in an influential textbook called the Cartan–Lie theorem the “third Lie theorem”, which became a rather popular term in recent years, though one should correctly call so just the theorem on local solvability of Maurer–Cartan equation.
The historical article of L. Maurer is
A MathOverflow entry about Maurer-Cartan forms for Lie groups: maurer-cartan-form
For literature on the third Lie theorem from the point of view of Maurer–Cartan equations, compare the following references:
Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces
N. Bourbaki, Lie algebras and lie groups, historical appendix
F. Engel, P. Heegaard, Sophus Lie Samlede Avhandliger (Collected works)
Andrey Lazarev, Maurer-Cartan moduli and models for function spaces, arxiv/1109.3715
Around def. 3.1 in