arXiv subjects

Christopher L. Rogers

Publications and source records attributed to Christopher L. Rogers.

At least 19 recordsLinked to original sources

arXiv2026

Lie's Third Theorem for Lie $\infty$-Algebras

We introduce the theory of local minimal models for Kan simplicial manifolds, which provide the appropriate generalization of minimal Kan simplicial sets to geometric contexts. We use this to obtain the first proof of Lie's third theorem for finite-type Lie $\infty$-algebras: Every finite-type, homologically and non-negatively graded $L_\infty$-algebra over $\mathbb{R}$ integrates to a finite-dimensional Lie $\infty$-group. As a corollary, our construction yields a new explicit finite-dimensional model for the string Lie 2-group.

math.RA↗

arXiv2025

Higher differentiation via higher formal groupoids

We solve the differentiation problem for Lie $\infty$-groups. Our approach builds on a classical version of Cartier duality which canonically identifies the Hopf algebra of point distributions supported at the identity of a Lie group with the universal enveloping algebra of its Lie algebra. Hence, for Lie $\infty$-groups, we consider simplicial coalgebras of point distributions. To do this properly, we first develop the homotopy theory of pointed formal $\infty$-groupoids within K. Behrend and E. Getzler's framework for higher geometric stacks. These objects are "higher" but not "derived", which is an important distinction for the geometric applications in mind. The second part of our construction relies on a careful analysis of J. Pridham's variation of the Dold-Kan adjunction for cosimplicial algebras. Our main result is a differentiation functor at the level of 1-categories from finite-dimensional Lie $\infty$-groups to finite-type Lie $\infty$-algebras that is homotopically well-behaved, coordinate-free, and explicit yet tractable. In particular, if $G_\bullet$ is a simplicial Lie group with Lie algebra $\mathfrak{g}_\bullet$, we prove that the differentiation of its classifying space $\overline{\mathscr{W}}_{\! \bullet}G$ is canonically isomorphic to the dg Lie algebra of normalized chains $N_\ast(\mathfrak{g}_\bullet)$.

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arXiv2023

On the Goldman-Millson theorem for $A_\infty$-algebras in arbitrary characteristic

Complete filtered $A_\infty$-algebras model certain deformation problems in the noncommutative setting. The formal deformation theory of a group representation is a classical example. With such applications in mind, we provide the $A_\infty$ analogs of several key theorems from the Maurer-Cartan theory for $L_\infty$-algebras. In contrast with the $L_\infty$ case, our results hold over a field of arbitrary characteristic. We first leverage some abstract homotopical algebra to give a concise proof of the $A_\infty$-Goldman-Millson theorem: The nerve functor, which assigns a simplicial set $\mathcal{N}_{\bullet}(A)$ to an $A_\infty$-algebra $A$, sends filtered quasi-isomorphisms to homotopy equivalences. We then characterize the homotopy groups of $\mathcal{N}_\bullet(A)$ in terms of the cohomology algebra $H(A)$, and its group of quasi-invertible elements. Finally, we return to the characteristic zero case and show that the nerve of $A$ is homotopy equivalent to the simplicial Maurer-Cartan set of its commutator $L_\infty$-algebra. This answers a question posed by N. de Kleijn and F. Wierstra in arXiv:1809.07743.

math.QA↗

arXiv2023

Complete $L_\infty$-algebras and their homotopy theory

We analyze a model for the homotopy theory of complete filtered $L_\infty$-algebras intended for applications in algebraic and algebro-geometric deformation theory. We provide an explicit proof of an unpublished result of E.\ Getzler which states that the category $\hat{\mathsf{Lie}}_\infty$ of such $L_\infty$-algebras and filtration-preserving $\infty$-morphisms admits the structure of a category of fibrant objects (CFO) for a homotopy theory. Novel applications of our approach include explicit models for homotopy pullbacks, and an analog of Whitehead's Theorem: under some mild conditions, every filtered $L_\infty$-quasi-isomorphism in $\hat{\mathsf{Lie}}_\infty$ has a filtration preserving homotopy inverse. Also, we show that the simplicial Maurer--Cartan functor, which assigns a Kan simplicial set to each $L_\infty$-algebra in $\hat{\mathsf{Lie}}_\infty$, is an exact functor between the respective CFOs. Finally, we provide an obstruction theory for the general problem of lifting a Maurer-Cartan element through an $\infty$-morphism. The obstruction classes reside in the associated graded mapping cone of the corresponding tangent map.

math.AT↗

arXiv2021

Which homotopy algebras come from transfer?

We characterize $A_\infty$-structures that are transfers over a chain homotopy equivalence or a quasi-isomorphism, answering a question posed by D. Sullivan. Along the way, we present an obstruction theory for weak $A_\infty$-morphisms over an arbitrary commutative ring. We then generalize our results to ${\mathcal P}_\infty$-structures over a field of characteristic zero, for any quadratic Koszul operad ${\mathcal P}$.

math.AT↗

arXiv2019

An explicit model for the homotopy theory of finite type Lie $n$-algebras

Lie $n$-algebras are the $L_\infty$ analogs of chain Lie algebras from rational homotopy theory. Henriques showed that finite type Lie $n$-algebras can be integrated to produce certain simplicial Banach manifolds, known as Lie $\infty$-groups, via a smooth analog of Sullivan's realization functor. In this paper, we provide an explicit proof that the category of finite type Lie $n$-algebras and (weak) $L_\infty$-morphisms admits the structure of a category of fibrant objects (CFO) for a homotopy theory. Roughly speaking, this CFO structure can be thought of as the transfer of the classical projective CFO structure on non-negatively graded chain complexes via the tangent functor. In particular, the weak equivalences are precisely the $L_\infty$ quasi-isomorphisms. Along the way, we give explicit constructions for pullbacks and factorizations of $L_\infty$-morphisms between finite type Lie $n$-algebras. We also analyze Postnikov towers and Maurer-Cartan/deformation functors associated to such Lie $n$-algebras. The main application of this work is our joint paper arXiv:1609.01394 with C. Zhu which characterizes the compatibility of Henriques' integration functor with the homotopy theory of Lie $n$-algebras and that of Lie $\infty$-groups.

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arXiv2019

On the homotopy theory for Lie $\infty$-groupoids, with an application to integrating $L_\infty$-algebras

Lie $\infty$-groupoids are simplicial Banach manifolds that satisfy an analog of the Kan condition for simplicial sets. An explicit construction of Henriques produces certain Lie $\infty$-groupoids called `Lie $\infty$-groups' by integrating finite type Lie $n$-algebras. In order to study the compatibility between this integration procedure and the homotopy theory of Lie $n$-algebras introduced in the companion paper arXiv:1809.05999, we present a homotopy theory for Lie $\infty$-groupoids. Unlike Kan simplicial sets and the higher geometric groupoids of Behrend and Getzler, Lie $\infty$-groupoids do not form a category of fibrant objects (CFO), since the category of manifolds lacks pullbacks. Instead, we show that Lie $\infty$-groupoids form an `incomplete category of fibrant objects' in which the weak equivalences correspond to `stalkwise' weak equivalences of simplicial sheaves. This homotopical structure enjoys many of the same properties as a CFO, such as having, in the presence of functorial path objects, a convenient realization of its simplicial localization. We further prove that the acyclic fibrations are precisely the hypercovers, which implies that many of Behrend and Getzler's results also hold in this more general context. As an application, we show that Henriques' integration functor is an exact functor with respect to a class of distinguished fibrations which we call `quasi-split fibrations'. Such fibrations include acyclic fibrations as well as fibrations that arise in string-like extensions. In particular, integration sends $L_\infty$ quasi-isomorphisms to weak equivalences, quasi-split fibrations to Kan fibrations, and preserves acyclic fibrations, as well as pullbacks of acyclic/quasi-split fibrations.

math.AT↗

arXiv2017

The cohomology of the full directed graph complex

In his seminal paper "Formality conjecture", M. Kontsevich introduced a graph complex $GC_{1ve}$ closely connected with the problem of constructing a formality quasi-isomorphism for Hochschild cochains. In this paper, we express the cohomology of the full directed graph complex explicitly in terms of the cohomology of $GC_{1ve}$. Applications of our results include a recent work by the first author which completely characterizes homotopy classes of formality quasi-isomorphisms for Hochschild cochains in the stable setting.

math.KT↗

arXiv2016

Homotopical properties of the simplicial Maurer-Cartan functor

We consider the category whose objects are filtered, or complete, $L_\infty$-algebras and whose morphisms are $\infty$-morphisms which respect the filtrations. We then discuss the homotopical properties of the Getzler-Hinich simplicial Maurer-Cartan functor which associates to each filtered $L_\infty$-algebra a Kan simplicial set, or $\infty$-groupoid. In previous work with V. Dolgushev, we showed that this functor sends weak equivalences of filtered $L_\infty$-algebras to weak homotopy equivalences of simplicial sets. Here we sketch a proof of the fact that this functor also sends fibrations to Kan fibrations. To the best of our knowledge, only special cases of this result have previously appeared in the literature. As an application, we show how these facts concerning the simplicial Maurer--Cartan functor provide a simple $\infty$-categorical formulation of the Homotopy Transfer Theorem.

math.AT↗

arXiv2016

Homotopy moment maps

Associated to any manifold equipped with a closed form of degree >1 is an `L-infinity algebra of observables' which acts as a higher/homotopy analog of the Poisson algebra of functions on a symplectic manifold. In order to study Lie group actions on these manifolds, we introduce a theory of homotopy moment maps. Such a map is a L-infinity morphism from the Lie algebra of the group into the observables which lifts the infinitesimal action. We establish the relationship between homotopy moment maps and equivariant de Rham cohomology, and analyze the obstruction theory for the existence of such maps. This allows us to easily and explicitly construct a large number of examples. These include results concerning group actions on loop spaces and moduli spaces of flat connections. Relationships are also established with previous work by others in classical field theory, algebroid theory, and dg geometry. Furthermore, we use our theory to geometrically construct various L-infinity algebras as higher central extensions of Lie algebras, in analogy with Kostant's quantization theory. In particular, the so-called `string Lie 2-algebra' arises this way.

math.DG↗

arXiv2016

Higher U(1)-gerbe connections in geometric prequantization

We promote geometric prequantization to higher geometry (higher stacks), where a prequantization is given by a higher principal connection (a higher gerbe with connection). We show fairly generally how there is canonically a tower of higher gauge groupoids and Courant groupoids assigned to a higher prequantization, and establish the corresponding Atiyah sequence as an integrated Kostant-Souriau infinity-group extension of higher Hamiltonian symplectomorphisms by higher quantomorphisms. We also exhibit the infinity-group cocycle which classifies this extension and discuss how its restrictions along Hamiltonian infinity-actions yield higher Heisenberg cocycles. In the special case of higher differential geometry over smooth manifolds we find the L-infinity-algebra extension of Hamiltonian vector fields -- which is the higher Poisson bracket of local observables -- and show that it is equivalent to the construction proposed by the second author in n-plectic geometry. Finally we indicate a list of examples of applications of higher prequantization in the extended geometric quantization of local quantum field theories and specifically in string geometry.

math-ph↗

arXiv2016

On an enhancement of the category of shifted L-infinity algebras

We construct a symmetric monoidal category $LIE^{MC}$ whose objects are shifted L-infinity algebras equipped with a complete descending filtration. Morphisms of this category are "enhanced" infinity morphisms between shifted L-infinity algebras. We prove that any category enriched over $LIE^{MC}$ can be integrated to a simplicial category whose mapping spaces are Kan complexes. The advantage gained by using enhanced morphisms is that we can see much more of the simplicial world from the L-infinity algebra point of view. We use this construction in a subsequent paper to produce a simplicial model of a $(\infty,1)$-category whose objects are homotopy algebras of a fixed type.

math.CT↗

arXiv2015

A Version of the Goldman-Millson Theorem for Filtered L-infinity Algebras

In this paper we consider $L_{\infty}$-algebras equipped with complete descending filtrations. We prove that, under some mild conditions, an $L_{\infty}$ quasi-isomorphism $U: L \to \tilde{L}$ induces a weak equivalence between the Deligne-Getzler-Hinich (DGH) $\infty$-groupoids corresponding to $L$ and $\tilde{L}$, respectively. This paper may be considered as a modest addition to foundational paper arXiv:math/0404003 by Ezra Getzler.

math.AT↗

arXiv2015

What do homotopy algebras form?

In paper arXiv:1406.1744, we constructed a symmetric monoidal category $LIE^{MC}$ whose objects are shifted (and filtered) L-infinity algebras. Here, we fix a cooperad $C$ and show that algebras over the operad $Cobar(C)$ naturally form a category enriched over $LIE^{MC}$. Following arXiv:1406.1744, we "integrate" this $LIE^{MC}$-enriched category to a simplicial category $HoAlg^Δ_C$ whose mapping spaces are Kan complexes. The simplicial category $HoAlg^Δ_C$ gives us a particularly nice model of an $(\infty,1)$-category of $Cobar(C)$-algebras. We show that the homotopy category of $HoAlg^Δ_C$ is the localization of the category of $Cobar(C)$-algebras and infinity morphisms with respect to infinity quasi-isomorphisms. Finally, we show that the Homotopy Transfer Theorem is a simple consequence of the Goldman-Millson theorem.

math.CT↗

arXiv2015

Kontsevich's graph complex, GRT, and the deformation complex of the sheaf of polyvector fields

We generalize Kontsevich's construction of L-infinity derivations of polyvector fields from the affine space to an arbitrary smooth algebraic variety. More precisely, we construct a map (in the homotopy category) from Kontsevich's graph complex to the deformation complex of the sheaf of polyvector fields on a smooth algebraic variety. We show that the action of Deligne-Drinfeld elements of the Grothendieck-Teichmueller Lie algebra on the cohomology of the sheaf of polyvector fields coincides with the action of odd components of the Chern character. Using this result, we deduce that the A-hat genus in the Calaque-Van den Bergh formula arXiv:0708.2725 for the isomorphism between harmonic and Hochschild structures can be replaced by a generalized A-hat genus.

math.KT↗

arXiv2014

L-infinity algebras of local observables from higher prequantum bundles

To any manifold equipped with a higher degree closed form, one can associate an L-infinity algebra of local observables that generalizes the Poisson algebra of a symplectic manifold. Here, by means of an explicit homotopy equivalence, we interpret this L-infinity algebra in terms of infinitesimal autoequivalences of higher prequantum bundles. By truncating the connection data on the prequantum bundle, we produce analogues of the (higher) Lie algebras of sections of the Atiyah Lie algebroid and of the Courant Lie 2-algebroid. We also exhibit the L-infinity cocycle that realizes the L-infinity algebra of local observables as a Kirillov-Kostant-Souriau-type L-infinity extension of the Hamiltonian vector fields. When restricted along a Lie algebra action, this yields Heisenberg-like L-infinity algebras such as the string Lie 2-algebra of a semisimple Lie algebra.

math-ph↗

arXiv2012

Notes on Algebraic Operads, Graph Complexes, and Willwacher's Construction

We give a detailed proof of T. Willwacher's theorem arXiv:1009.1654 which links the cohomology of the full graph complex fGC to the cohomology of the deformation complex of the operad GER, governing Gerstenhaber algebras. We also present various prerequisites required for understanding the material of arXiv:1009.1654. In particular, we review operads, cooperads, and the cobar construction. We give a detailed exposition of the convolution Lie algebra and its properties. We prove a useful lifting property for maps from a dg operad obtained via the cobar construction. We describe in detail Willwacher's twisting construction, and then use it to work with various operads assembled from graphs, in particular, the full graph complex and its subcomplexes. These notes are loosely based on lectures given by the first author at the Graduate and Postdoc Summer School at the Center for Mathematics at Notre Dame (May 31 - June 4, 2011).

math.KT↗

arXiv2012

2-plectic geometry, Courant algebroids, and categorified prequantization

A 2-plectic manifold is a manifold equipped with a closed nondegenerate 3-form, just as a symplectic manifold is equipped with a closed nondegenerate 2-form. In 2-plectic geometry we meet higher analogues of many structures familiar from symplectic geometry. For example, any 2-plectic manifold has a Lie 2-algebra consisting of smooth functions and Hamiltonian 1-forms. This is equipped with a Poisson-like bracket which only satisfies the Jacobi identity up to `coherent chain homotopy'. Over any 2-plectic manifold is a vector bundle equipped with extra structure called an exact Courant algebroid. This Courant algebroid is the 2-plectic analogue of a transitive Lie algebroid over a symplectic manifold. Its space of global sections also forms a Lie 2-algebra. We show that this Lie 2-algebra contains an important sub-Lie 2-algebra which is isomorphic to the Lie 2-algebra of Hamiltonian 1-forms. Furthermore, we prove that it is quasi-isomorphic to a central extension of the (trivial) Lie 2-algebra of Hamiltonian vector fields, and therefore is the higher analogue of the well-known Kostant-Souriau central extension in symplectic geometry. We interpret all of these results within the context of a categorified prequantization procedure for 2-plectic manifolds. In doing so, we describe how U(1)-gerbes, equipped with a connection and curving, and Courant algebroids are the 2-plectic analogues of principal U(1) bundles equipped with a connection and their associated Atiyah Lie algebroids.

math-ph↗