# nLab homomorphism of L-∞ algebras

Contents

### Context

#### $\infty$-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

$\infty$-Lie groupoids

$\infty$-Lie groups

$\infty$-Lie algebroids

$\infty$-Lie algebras

# Contents

## Idea

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

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

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

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

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

• “strong homotopy maps”, abbreviated: “sh maps”

• $L_\infty$-morphisms”