∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
-Lie groupoids
-Lie groups
-Lie algebroids
-Lie algebras
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
There are two notions of homomorphism of -algebras, one a special case of the other, depending on the homotopy theoretic perspective:
If an -algebra is regarded as a chain complex equipped with higher-ary-bracket operations , then a homomorphism is a chain map which is compatible with the bracket operations.
However, even when so regarded, L-∞ algebras are objects of a homotopical category, in fact of the category of fibrant objects of a model structure for -algebras, so that the homotopy-correct morphisms out of are those out of a cofibrant resolution , hence are zig-zags of naive morphisms, as above, of the form .
But such poperly resolved morphisms are equivalently just the evident homomorphism of the associated Chevalley-Eilenberg dg-coalgebras (here), dually of the CE-dg-algebras (here):
These homotopy-correct homomorphisms of -algebras are known under a variety of different names, including:
“strong homotopy maps”, abbreviated: “sh maps”
(Stasheff following Sugawara 60, Sec. 2, see also Merkulov 02, 2.11),
“weak maps”
“-morphisms”
Tom Lada, Martin Markl, Rem. 5.3 of: Strongly homotopy Lie algebras, Communications in Algebra Volume 23, Issue 6, (1995) (arXiv:hep-th/9406095)
Maxim Kontsevich, Deformation quantization of Poisson manifolds, I, Lett. Math. Phys. 66 (2003) 157-216 (arXiv:q-alg/9709040, doi:10.1023/B:MATH.0000027508.00421.bf)
Sergei A. Merkulov, Operads, deformation theory and -manifolds (arXiv:math/0210478)
The terminology “strong homotopy” for the homotopy-coherent morphisms was originally borrowed from:
Last revised on July 23, 2021 at 08:17:28. See the history of this page for a list of all contributions to it.