on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
∞-Lie theory (higher geometry)
There exist various model category structures which present the homotopy theory of L-∞ algebras.
By definition L-∞ algebras are the ∞-algebras in the category of chain complexes over the Lie operad. As such they carry a model structure on algebras over an operad. There is a strictification which leads equivalently to a model structure on dg-Lie algebras.
A more geometric way is to think of L-∞ algebras as being the tangent spaces to connected smooth ∞-groupoids, hence to the delooping/moduli ∞-stacks $\mathbf{B}G$ of smooth ∞-groups, hence as the first order infinitesimal neighbourhood
of the essentially unique point $* \to \mathbf{B}G$ (see at Lie differentiation). Since this is equivalently the first order neighbourhood of the formal neighbourhood (the jets) and the formal neighbourhood can be described purely in terms of associative algebra/coalgebra (see at smooth algebra) many models for $L_\infty$-algebras are formulated in terms of such data.
In particular, one succinct way to present L-∞ algebras (as discussed there) is as dg-coalgebras:
An L-∞ algebra $(\mathfrak{g}, [-], [-,-], [-,-,-], \cdots)$ structure on a graded vector space $\mathfrak{g}$ is equivalently a dg-coalgebra structure on the graded-commutative cofree coalgebra over $\mathfrak{g}$.
Conversely, the category of L-∞ algebras (and general “weak” morphisms between them) is the full subcategory of that of counital cocommutative dg-coalgebras on those whose underlying bare graded-commutative coalgebra (forgetting the codifferential) is free
Accordingly it is of interest to have also model structure on dg-coalgebras or dually (on pro-objects of) dg-algebras which presents $L_\infty$-algebras. Prop. 9 below identifies the category $L_\infty Alg$ as the (full sub-)category of fibrant objects inside such a model structure of “differential graded formal spaces”, which in turn is related by a zig-zag of Quillen equivalence to various other models
So we write “$L_\infty Alg$” here for the 1-category of $L_\infty$-algebras and general morphisms between them, since this is an entry on model category presentations. If we want to refer to the (∞,1)-category of $L_\infty$-algebras we here write explicitly “$L_W( L_\infty Alg)$”, referring to the simplicial localization of this 1-category.
All gradings in the following are $\mathbb{Z}$-gradings, unless explicitly stated otherwise. In terms of the underlying geometry this means that we are dealing with derived geometry (see below the section Simplicial sheaves over comsimplicial formal spaces for details): the algebra elements in positive degree correspond to categorical/simplicial/∞-groupoid/∞-stack-degree, and those in negative degree to the cosimplicial degree of the derived site of cosimplicial formal spaces.
Technically this affects for instance the nature of fibrations: for instance the model structure on dg-Lie algebras below is transferred from a model structure on chain complexes. For unbounded chain complexes this is the “Categorical projective class structure” whose fibrations are the chain maps that are surjective in every degree. This appears for instance in prop. 2 and prop. 10 below.
On the other hand, if one considered chain complexes in non-negative degree (for tangent complexes in “higher but non-derived geometry”), then one would use the Projective structure on chain complexes in non-negative degree. This has as fibrations precisely the chain maps that are surjective in every positive degree. This case is (currently) not discussed in the following.
Some of the model structures below are on the category of $L_\infty$-algebras with “strict” morphisms between them, namely for those morphisms which are morphisms of algebras over an operad for an $L_\infty$-algebra regarded as an algebra over a cofibrant resolution of the Lie operad. We write
for the wide subcategory on the strict $L_\infty$-morphisms.
As their name indicates, L-∞ algebras are the homotopy algebras over the Lie operad (in a category of chain complexes). As such, the general theory of model structures on algebras over an operad provides a model category structure on the category of $L_\infty$-algebras.
This we discuss here. But there is also a natural identification of $L_\infty$-algebras with infinitesimal derived ∞-stacks. For expressing this a host of other, Quillen equivalent model structures are available. These we discuss below in Definitions as formal/infinitesimal ∞-stacks.
By the general discussion at model structure on dg-algebras over an operad, if $k$ is a field which contains the field of rational numbers, then for every symmetric operad (uncolored) $\mathcal{O}$ in the category of chain complexes (unbounded) $Ch_\bullet(k)$, the free-forgetful adjunction
between the algebras over an operad and the underlying chain complexes induces a transferred model structure from the projective unbounded model structure on chain complexes where hence on both sides
the weak equivalences are the morphisms that are quasi-isomorphisms on the (underlying) chain complexes;
the fibrations are the morphisms that are degreewise surjections on the (underlying) chain complexes.
Hence in particular $(F \vdash U)$ is Quillen adjunction between these model structures.
So this is in particular true for $\mathcal{O} = \widehat Lie$ the standard cofibrant resolution of the Lie operad. In this case $Alg(\widehat Lie) \simeq L_\infty Alg_{str}$ is the category of (unbounded) $L_\infty$-algebras (with strict $L_\infty$-maps between them as in remark 3 above) and hence is equipped with a transferred model structure this way
Moreover, by the rectification result discussed at model structure on dg-algebras over an operad, the resolution map $\widehat Lie \stackrel{\simeq}{\to} Lie$ induces a Quillen equivalence
with the model structure on dg-Lie algebras, similarly transferred from the model structure on chain complexes.
We list here definitions of various further model category structures that all present the (∞,1)-category of L-∞ algebras and describe a web of zig-zags of Quillen equivalences between them. These Quillen equivalences may be thought of as presenting an equivalence between the (∞,1)-category of $L_\infty$-algebras and that of infinitesimal derived ∞-stacks (“formal moduli problems”).
The following tabulates the main categories considered below, the functors relating them and their homotopy theoretic nature. The last row points to the relevant definitions and propositions of the following text.
L-∞ algebras | form Chevalley-Eilenberg algebra | pro-objects in commutative Artin dg-algebras | dualize | commutative dg-coalgebra | form tangents | dg-Lie algebras |
---|---|---|---|---|---|---|
$L_\infty Alg$ | $\stackrel{CE}{\hookrightarrow}$ | $Pro(dgArtinCAlg)^{op}$ | $\stackrel{(-)^*}{\hookrightarrow}$ | $dgCoCAlg$ | $\stackrel{\mathcal{L}}{\to}$ | $dgLieAlg$ |
$=: dgFormalSpace$ | ||||||
category of fibrant objects | equivalence of (∞,1)-categories under simplicial localization | opposite model structure of cofibrantly generated model category | left Quillen equivalence | model category | left Quillen equivalence | cofibrantly generated model category |
prop. 1 | def. 9 | def. 8 | prop. 15 | def. 3 | prop. 3 | def. 1 |
Here we are trying to use suggestive names of the categories involved. The notation used here corresponds to that in (Pridham) by the following dictionary
(handle with care, may still need attention)
notation used here | notation in Pridham |
---|---|
$DerivedFormalSpace$, def. 6 | $scSp$, def. 1.32 |
$dgFormalSpace$, def. 8 | $DG_\mathbb{Z}Sp$. def. 3.1 |
$FormalSpace^{\Delta^{op}}$ | $sDGSp$, def. 4.6 |
Let $k$ be a field of characteristic 0.
Write $dgLieAlg_k \in Cat$ for the category of dg-Lie algebras over $k$.
The category $dgLieAlg_k$ carries a cofibrantly generated model category structure in which
fibrations are the degreewise surjective maps;
weak equivalences are the quasi-isomorphisms
on the underlying chain complexes. This is the transferred model structure of the corresponding model structure on chain complexes along the forgetful functor to the category of chain complexes.
We call this the model structure on dg-Lie algebras.
Write
for the functor which sends a dg-Lie algebra $(\mathfrak{g},d,[-,-])$ to the dg-coalgebra whose underlying coalgebra is free on the underlying graded vector space $\mathfrak{g}$ and whose coderivation is given by
and then extended as a coderivation.
(The Chevalley-Eilenberg dg-coalgebra.)
The functor from def. 2 has a left adjoint
(Quillen, App. B6) (Hinich98, 1.2.1, 2.2.5) See also (Pridham, def. 3.23).
If one thinks of a dg-coalgebra as presenting a a derived formal space, as discuss below then its image under $\mathcal{L}$, prop. 3, may be thought of as its tangent dg-Lie algebra. Therefore $\mathcal{L}$ is also called the tangent Lie algebra functor.
Let $k$ be a field of characteristic 0.
Write $dgCoCAlg_k \in Cat$ for the category of co-commutative counital $\mathbb{Z}$-graded dg-coalgebras over $k$.
There exists a model category structure on $dgCoCAlg_k$ for which
the cofibrations are the morphisms that are degreewise injections;
the weak equivalences are the morphisms $f$ whose corresponding morphisms of dg-Lie algebras $\mathcal{L}(f)$, prop. 3, is a quasi-isomorphism on the underlying chain complexes.
(Hinich98, theorem 3.1) See also (Pridham, lemma 3.25).
We call this the model structure on dg-coalgebras.
Beware that the class of weak equivalences in prop. 4 is not that of quasi-isomorphisms on the chain complexes underlying the dg-coalgebras.
But they form a sub-class:
The Chevalley-Eilenberg functor
from def. 2 sends quasi-isomorphisms on the chain complexes underlying dg-Lie algebras to quasi-isomorphisms on the chain complexes underlying their Chevalley-Eilenberg dg-coalgebras.
The pair of adjoint functors
from prop. 3 constitutes a Quillen equivalence between the model structure on dg-Lie algebras, prop. 2, and the model structure on dg-coalgebras, prop. 4.
Since every object in $dgCoCAlg_k$ is a cofibrant object and every object in $dgLie_k$ is a fibrant object, the composite
is already its derived functor and the unit
is a weak equivalence that exhibits $\mathcal{C}\mathcal{L}\mathfrak{g}$ as a fibrant resolution and moreover, if $\mathfrak{g}$ was already fibrant, hence by prop. 9 below an L-∞ algebra, as a strictification of $\mathfrak{g}$: because a dg-Lie algebra is an $L_\infty$-algebra in which the Lie bracket satisfies its Jacobi identity strictly (not just up to a homotopy measured by the trinary bracket) and in which the “Jacobiator identity” holds strictly, etc.
Write
for the category of infinitesimally thickened points, the full subcategory of the opposite category of Artin algebras (“Weil alebras” in the language of synthetic differential geometry). The category of infinitesimally thickened points.
Write
for the full subcategory of the opposite category on simplicial algebras on those which are Artinian (or “Weil” ): the category of cosimplicial infinitesimally thickened points.
Write
for the full subcategory of the category of presheaves over infinitesimally thickened points on those given by left exact functors.
In (Pridham) this is def. 1.18.
Write
for the full subcategory of the category of simplicial presheaves over cosimplicial infinitesimally thickened points on those given by left exact functors.
In (Pridham) this is def. 1.32.
There exists a cofibrantly generated model category structure on $DerivedFormalSpace$ whose
(…)
This is (Pridham, def. 2.7, theorem 2.14)
Between quasi-smooth objects in $DerivedFormalSpace$ the weak equivalences are precisely the morphisms which are weak homotopy equivalences of simplicial sets over each object in $cInfThPoint$.
This is (Pridham, cor. 2.16).
Let $\Lambda$ be a commutative ring which is
local with
maximal ideal $\mu$
residue field $k$.
Write
for the category of commutative local Artin algebras over $\Lambda$ whose residue field is also $k$.
If $\mu = 0$ then $\Lambda = k$ and we just write
In all of the following this is assumed to be the case, with $k$ of characteristic zero.
Write
for the category of $\mathbb{Z}$-graded graded-commutative Artin dg-algebras over $k$ with residue field $k$.
Write
for its category of pro-objects and write
for the opposite category of that.
This is (Pridham, def. 3.1) following (Manetti 02).
While it so happens that every coalgebra and dg-coalgebra is the filtered colimit of its finite-dimensional subalgebras (see at coalgebra – As filtered colimits), this is not in general the case for algebras. But it follows that the linear dual of a general coalgebra is a filtered limit of finite-dimensional algebras, hence a pro-object in finite dimensional algebras. This is the reason for the appearance of pro-objects in def. 8.
There is a cofibrantly generated model category structure on $Pro(dgArtinCAlg_k)$, def. 8 – hence an opposite model structure on $dgFormalSpaces$ – whose weak equivalences are those morphisms that are local morphisms relative to quasi-smooth maps in the homotopy category of the slice category over their codomain.
This is (Pridham, prop. 4.36).
Write
for the functor which regards an L-infinity algebra $\mathfrak{g}$ as a dg-coalgebra by prop. 1 and then forms the linear dual dg-algebra, the Chevalley-Eilenberg algebra $CE(\mathfrak{g})$ of $\mathfrak{g}$ (a pro-dg-algebra according to spring).
$L_\infty$-algebras are precisely the fibrant objects in $dgFormalSpace$: the Chevalley-Eilenberg algebra functor of def. 9,
is an equivalence of categories onto its essential image, which are the fibrant objects of $dgFormalSpace$:
This is proven inside the proof of (Pridham, prop. 4.42).
Prop. 9 shows in particular that the category $L_\infty Alg$ of prop/def. 1 carries the structure of a category of fibrant objects that presents the homotopy theory of $L_\infty$-algebras. Notice that, of course, passing to the full subcategory of fibrant objects does not change the homotopy theory presented by the underlying category with weak equivalences in that we have an equivalence of (∞,1)-categories between the simplicial localizations
The following proposition characterizes the structure of this category of fibrant objects.
The induced structure of a category of fibrant objects on $L_\infty Alg$ under the inclusion of prop. 9 has
weak equivalences are precisely the maps that are quasi-isomorphisms on the underlying chain complexes;
fibrations include in particular the maps that are surjections on the underlying chain complexes.
The first statement is proven in the proof of (Pridham, prop. 4.42).
The second statement follows by (Pridham, def. 4.34) with the existence of the model structure on $dgFormalSpaces$.
Should spell out how this follows, using lifting.
Beware, as in remark 5, that the class of weak equivalences in prop. 10 differs from that of those maps on associated Chevalley-Eilenberg algebras which are quasi-isos on the underlying chain complexes of the dg-algebra (which instead are the weak equivalences in the standard model structure on dg-algebras, hence in particular those used in Sullivan rational homotopy theory). Instead the weak equivalences correspond to the maps of CE-algebra that are quasi-isomorphisms only on the chain complexes given by the co-unary component of the differential of the CE-algebra.
There is an equivalence of categories between the homotopy category of $dgFormalSpace$ hence of $L_\infty Alg$ according to remark 8, and the homotopy category of $L_\infty$-algebras according to (Kontsevich 94).
(Pridham, prop. 4.42, see above def. 4.29)
If $k$ is of characteristic 0 then there is a zig-zag of Quillen equivalences between $DerivedFormalSpace$, def. 6 and $dgFormalSpace$, def. 8, hence an equivalence of (∞,1)-categories between their simplicial localizations
This is (Pridham, cor. 4.49).
For arbitrary $k$, there is a Quillen equivalence
The inclusion
given by sending an object in $Pro(dgArticCAlg_k)^{op} \coloneqq dgFormalSpace$, hence an dg-algebra $A$, to its dual dg-coalgebra $A^*$, is the left adjoint part of a Quillen equivalence between the model structure on $dgFormalSpace$, prop. 8, and the model structure on dg-coalgebras, prop. 4.
Also a version of the “dual monoidal Dold-Kan correspondence” gives a Quillen equivalence between two model structures for $L_\infty$-algebras. This is (Pridham, section 4.4). This we discuss now
This equivalence has the nice property that starting with the Chevalley-Eilenberg algebra and then “denormalizing” it under dual monoidal Dold-Kan to a cosimplicial nilpotent algebra yields manifestly an incarnation of the $L_\infty$-algebra in terms of simplicial complexes of infinitesimal simplices as is implicit in the work of Anders Kock in synthetic differential geometry. This is spelled out further in dcct, section 4.5.1.
Write
for cosimplicial pro-objects of dg-Artin algebras ($\mathbb{N}$-graded).
The category $(dg\hat{\mathcal{C}})^{\Delta}$ of def. 10 carries a model category structure where
(…)
This is (Pridham, def. 4.11, prop. 4.12).
Write
for pro-objects in dg-algebras ($\mathbb{N}$-graded) in dg-Artin algebras ($\mathbb{N}$-graded).
The dual monoidal Dold-Kan correspondence functor from dg-algebras to cosimplicial algebras? (the inverse equivalence to the normalized cochain complex functor)
induces on $DGdg\hat \mathcal{C}$ the transferred model structure from that of prop. 16 and is the right adjoint of a Quillen equivalence with respect to these model structures
This is (Pridham, theorem 4.26).
We discuss some further properties of the above model category structures.
The model category $dgFormalSpace$, def. 8, is a right proper model category.
This observation has been communicated privately by Jonathan Pridham
We need to show that the pullback of a weak equivalence $w$ along a fibration $f$ is again a weak equivalence. If $w$ is a fibration, this is automatic, so by factorisation we reduce to the case where $w$ is a cofibration. Now, every trivial cofibration is $Spf$ of a composition of acyclic small extensions, so we may take $w$ to be $Spf$ of an acyclic small extension $A \to B$ with kernel $I$. Then $f$ is $Spf$ of a quasi-free map $A \to R$, so the pullback is $Spf$ of $R \to R/IR$, and $IR =I\hat{\otimes}_A R$, so $R \to R/IR$ is also an acyclic small extension.
In any model category we have a notion of homotopy between 1-morphisms. In any category of fibrant objects we still have a notion of right homotopy, given by maps into a path space object. So all of the above model category/fibrant object category structures yield models for homotopies between morphisms of $L_\infty$-algebras.
A discussion of path space objects of and hence of right homotopies between $L_\infty$-algebras (in the category of def. 1) is for instance in (Dolgushev 07, section 5).
More generally, a description of the full derived hom space between two $L_\infty$-algebras is obtained via remark 8 from the description of derived hom-spaces in categories of fibrant objects.
Recognizing homotopy fiber products in any of the model structure above can be a bit subtle. A recognition principle of homotopy fibers over abelian $L_\infty$-algebras , hence useful for discussion of ∞-Lie algebra extensions), is described in (Fiorenza-Rogers-Schreiber 13, theorem 3.1.13).
Precursors for 2-reduced dg-algebras are dicussed in
The homotopy-theoretic nature of $L_\infty$-algebras and their relation to deformation problems was then notably amplified in
Model structurs on algebras over operads in chain complexes were discussed generally in
The full model structure on dg-coalgebras (in characteristic 0) and the Quillen equivalence of dg-Lie algebras as well as the interpretation in terms of formal $\infty$-stacks is due to
In
the relation to $\infty$-stacks is discussed more in detail.
More model category theoretic developments relating various of the previous approaches and generalizing to arbitrary characteristic are in
in parts based on
A useful summary of that paper is given in the notes, by Stefano Maggiolo.
A discussion of path space objects for $L_\infty$-algebras is in section 5 of
A discussion of homotopy fibers of morphusms to abelian $L_\infty$-algebras and hence ∞-Lie algebra extensions) is in section 3.1 of