nLab
homomorphism of L-∞ algebras

Contents

Context

\infty-Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Homotopy theory

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

Contents

Idea

There are two notions of homomorphism of L L_\infty -algebras, one a special case of the other, depending on the homotopy theoretic perspective:

If an L L_\infty -algebra 𝔤=(V ,[])\mathfrak{g} = (V_\bullet, [\cdots]) is regarded as a chain complex V V_\bullet equipped with higher-ary-bracket operations [][\cdots], then a homomorphism 𝔤𝔤\mathfrak{g} \xrightarrow{\;\;} \mathfrak{g}' is a chain map V V V_\bullet \xrightarrow{\;\;} V'_\bullet 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 L L_\infty -algebras, so that the homotopy-correct morphisms out of 𝔤\mathfrak{g} are those out of a cofibrant resolution 𝔤^W𝔤\varnothing \xhookrightarrow{\;} \widehat {\mathfrak{g}} \xrightarrow{ \in \mathrm{W} } \mathfrak{g}, hence are zig-zags of naive morphisms, as above, of the form 𝔤W𝔤^𝔤\mathfrak{g} \xleftarrow{\in \mathrm{W}} \widehat {\mathfrak{g}} \xrightarrow{\;\;\;} \mathfrak{g}'.

But such poperly resolved morphisms are equivalently just the evident homomorphism of the associated Chevalley-Eilenberg dg-coalgebras CE ()CE_\bullet(-) (here), dually of the CE-dg-algebras CE ()CE^\bullet(-) (here):

CE (𝔤)CE (𝔤),CE (𝔤)CE (𝔤). CE_\bullet(\mathfrak{g}) \xrightarrow{\;\;\;\;} CE_\bullet(\mathfrak{g}') \,, \;\;\;\;\;\;\;\; CE^\bullet(\mathfrak{g}) \xleftarrow{\;\;\;\;} CE^\bullet(\mathfrak{g}') \,.

These homotopy-correct homomorphisms of L L_\infty-algebras are known under a variety of different names, including:

References

The terminology “strong homotopy” for the homotopy-coherent morphisms was originally borrowed from:

Last revised on July 23, 2021 at 04:17:28. See the history of this page for a list of all contributions to it.