What is a Homotopy between $L_\infty$-algebra morphisms II

Question

I would like to continue on question I asking, what is a homotopy between Lie infinity algebras, since I'm not satisfied in two directions:

1.) The naive approach to define a homotopy would be ('naive' in my opinion of course) the following:

Let $(L,(l)_{k\in\mathbb{N}})$ and $(M,(d)_{k\in\mathbb{N}})$ be two Lie infinity algebras, let $f_\infty,g_\infty:L \to M$ be two morphism (in the most general sense) and let $(C(L),Q_L)$ and $(C(M),Q_M)$ be the appropriate differential graded coalgebras with induced morphism $F,G:C(L)\to C(M)$.

Then a homotopy between $F$ and $G$ is a degree $+1$ map

$H:C(L)\to C(M)$ such that $F-G = HQ_L \pm Q_MH$

let signs and additional structure of $H$ (linear, coalgebra, ...) aside for a moment.

It is strange, however, that I never saw this approach in the literature. Is this definition of homotopy equivalent to the previously mentioned approaches in question I?

Now for what is more important:

2.) The homotopy theory of Lie infinity algebras as given by Urs Schreiber in question I is obtained by 'transferring' the homotopy theory of differential graded Lie algebras 'along' an adjunction

$F\dashv G$

($F:L_\infty Alg \to DGLA$ and $G:DGLA \to L_\infty Alg$)

Now the question that really irritates me for quite some time is: how can we be sure that we get the correct homotopy theory of Lie infinity algebras by transferring its 'shadow' in the category of DG Lie algebra back along the previous mentioned adjunction? To me it looks like we can not rule out that there is a more general definition of weak equivalences in the category of Lie infinity algebras, which just project under $F$ onto those we already know.

Sorry if the second question is vague.

Edit; I used the 'Lie algebra cohomology tag' since the homotopy theory of Lie infinity algebras affects Lie algebra cohomology, too.

Answer 1

1) Any choice of path (or cylinder) objects will give the same homotopy classes, using the model category structure. Unless M is abelian, these path objects are harder to construct than path objects for chain complexes (exercise). You can't just ignore the coalgebra structure.

2) Hinich's definition of weak equivalences of dg coalgebras was reflected from DGLAs, but for L-infinity algebras, these are equivalent to tangent quasi-isomorphisms. For general dg coalgebras, intrinsic and natural characterisations of weak equivalences exist, but are more involved. Note that these are stronger (not weaker, as the question seems to hope) than dg coalgebra quasi-isomorphisms.

Answer 2

First of all, you might want to look at the MO question How to define the equivalence of Maurer-Cartan elements in an $L_\infty$-algebra?, since of course this is effectively what you need (morphisms can be described as MC elements in a particular convolution $L_\infty$-algebra, I suspect you know that).

Let me also say that some sort of approach to homotopy of the kind you are asking for is discussed by Martin Markl in this paper, and in this paper it is checked that the sort of definition proposed by Martin can be obtained from a Sullivan-type approach to homotopies (outlined in an answer to your first question) using appropriate homotopy transfer theorem for homotopy cooperads.

It is worth remarking, in particular, that the definition of a "derivation homotopy" suggested in another answer to this question does not work: the crucial difference between $A_\infty$ and $L_\infty$ is that $A_\infty$ is a nonsymmetric operad, and therefore instead of the Sullivan's algebra of differential forms on the 1-simplex we can use the dg algebras of chains on the 1-simplex (which is noncommutative) - this gives the "derivation homotopy".

Answer 3

I did not know where to put this answer, whether here or in this thread. Please, consider it as a remark on a possible definition of $L_\infty$-homotopies. I would like to summarize some considerations on $L_\infty$-algebra homotopies using as prototype the homotopies of $A_\infty$ morphisms.

With $A$ and $B$ we denote $\mathbb Z$-graded vector spaces on a ground field of characteristic zero (for simplicity). In what follows, a graded vector space will be also called "graded object".

1. A little remark: the chain complexes case

If $A$ resp. $B$ are chain complexes, with differentials $d_A$, resp. $d_B$, and $f,g: A\rightarrow B$ are (chain complexes) morphisms then a homotopy $H: A\rightarrow B$ between them is a degree $-1$ morphism of graded objects s.t. $f-g = d_BH+Hd_A$. No extra condition on $H$ is needed, as $d_B (f-g) = (f-g) d_A$ is automatically satisfied.

2. $L_\infty$ case

Let $(A,D_A)$ resp. $(B,D_B)$ by $L_\infty$-algebras; with $D_A$, resp. $D_B$ we denote the squaring to zero, $+1$ degree coderivations on the free graded co-commutative coalgebras over $A$, resp. $B$. These latter are denoted by $(C(A),\Delta_A)$ resp. $(C(B),\Delta_B)$: the $\Delta$'s are the coproducts.

Let $F,G$ be $L_\infty$-morphisms from $A$ to $B$, i.e. differential graded coalgebra maps $F,G:(C(A),D_A,\Delta_A)\rightarrow (C(B),D_B,\Delta_B)$ satisfying the known compatibility conditions w.r.t. coderivations and coproducts.

Definition I An $(F,G)$-derivation $R$ of degree $k$ is a graded object map $R:C(A)\rightarrow C(B)$ of degree $k$ s.t.
$$\Delta_B H = (F\otimes H+ H\otimes G)\Delta_A.~~ (**) $$

We arrive at the main definition of this answer, i.e.

Definition II A homotopy $H$ between the $L_\infty$-algebras morphisms $F$ and $G$ is an $(F,G)$-derivation of degree $-1$ s.t. $F-G = D_BH+HD_A$.

The questions are: why do we impose the $(F,G)$-coderivation relation (**) ? Which consequences does it have? The (inevitably partial) answers to these questions are my motivation for such definition of $L_\infty$-homotopies. In summary:

• As $H$ is an $(F,G)$-coderivation, then it satisfies a lifting property and it is uniquely determined by its components $H_n: A^{\otimes n}\rightarrow B$. $H$ is lifted with the well-known formulae $H=\sum F_{\bullet}^{\otimes\bullet}\otimes H_{\bullet}\otimes G_{\bullet}^{\otimes\bullet}$.

• $D_BH+HD_A$ is an $(F,G)$-coderivation ($D_A$ and $D_B$ are $(1,1)$-coderivations) as well; this implies that the relation $F-G = D_BH+HD_A$ can be expressed with the familiar tower of $L_\infty$-relations found in the literature.

• $G + D_BH+HD_A$ and $F-D_BH+HD_A$ really cocommute with coproducts, as $L_\infty$-algebra morphisms should.

In summary, I believe a definition of homotopy of $L_\infty$-algebras could be the one given in definition II, above. As said in the introduction, similar constructions take place in the realm of $A_\infty$-algebras: in this setting one uses as graded coalgebras the reduced tensor coalgebras. A good reference can be the first chapter in Hasegawa's PhD Thesis.