nLab homomorphism of L-∞ algebras

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”

  (Stasheff following Sugawara 60, Sec. 2, see also Merkulov 02, 2.11),

• “weak maps”

  (Lada & Markl 1994, Rem. 5.3),

• “$L_\infty$-morphisms”

  (Kontsevich 1997, p. 12)

References

• 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 $F$-manifolds (arXiv:math/0210478)

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

• Masahiro Sugawara, On the homotopy-commutativity of groups and loop spaces, Mem. College Sci. Univ. Kyoto Ser. A Math. 33(2): 257-269 (1960) (DOI:10.1215/kjm/1250775911)