My mathematical research is at the interface between (algebraic) operad theory and rational homotopy theory. Algebraic operads provide a powerful framework that allows us to speak about types of algebras, such as associative, commutative, and Lie algebras, and to study relations between different types of algebras. Through their lens, I explore rational homotopy theory with the goal of reformulating and extending Quillen's approach using homotopy Lie algebras and higher Lie theory.
Daniel Robert-Nicoud and Felix Wierstra
Journal of Noncommutative Geometry 13(4):1435-1462, 2019.
In previous works by the authors, a bifunctor was associated to any operadic twisting morphism, taking a coalgebra over a cooperad and an algebra over an operad, and giving back the space of (graded) linear maps between them endowed with a homotopy Lie algebra structure. We build on this result by using a more general notion of ∞-morphism between (co)algebras over a (co)operad associated to a twisting morphism, and show that this bifunctor can be extended to take such ∞-morphisms in either one of its two slots. We also provide a counterexample proving that it cannot be coherently extended to accept ∞-morphisms in both slots simultaneously. We apply this theory to rational models for mapping spaces.
Algebraic & Geometric Topology 19(3):1453-1476, 2019.
The goal of this paper is to introduce a smaller, but equivalent version of the Deligne-Hinich-Getzler ∞-groupoid associated to a homotopy Lie algebra. In the case of differential graded Lie algebras, we represent it by a universal cosimplicial object.
Daniel Robert-Nicoud and Felix Wierstra
Homology, Homotopy and Applications 21(1):351-373, 2019.
Given a coalgebra C over a cooperad, and an algebra A over an operad, it is often possible to define a natural homotopy Lie algebra structure on hom(C,A), the space of linear maps between them, called the convolution algebra of C and A. In the present article, we use convolution algebras to define the deformation complex for infinity-morphisms of algebras over operads and coalgebras over cooperads. We also complete the study of the compatibility between convolution algebras and infinity-morphisms of algebras and coalgebras. We prove that the convolution algebra bifunctor can be extended to a bifunctor that accepts infinity-morphisms in both slots and which is well defined up to homotopy, and we generalize and take a new point of view on some other already known results. This paper concludes a series of works by the two authors dealing with the investigation of convolution algebras.
Graduate Journal of Mathematics 3(1):15-30, 2018.
By a result of Vallette, we put a sensible model structure on the category of conilpotent Lie coalgebras. This gives us a powerful tool to study the subcategory of Lie algebras obtained by linear dualization, also known as the category of pronilpotent Lie algebras. This way, we recover weaker versions of the celebrated Goldman-Millson theorem and Dolgushev-Rogers theorem by purely homotopical methods. We explore the relations of this procedure with the existent literature, namely the works of Lazarev-Markl and Buijs-Félix-Murillo-Tanré.
Documenta Mathematica 23:189-240, 2018.
In order to solve two problems in deformation theory, we establish natural structures of homotopy Lie algebras and of homotopy associative algebras on tensor products of algebras of different types and on mapping spaces between coalgebras and algebras. When considering tensor products, such algebraic structures extend the Lie algebra or associative algebra structures that can be obtained by means of the Manin products of operads. These new homotopy algebra structures are proven by to compatible with the concepts of homotopy theory: ∞-morphisms and the Homotopy Transfer Theorem. We give a conceptual interpretation of their Maurer-Cartan elements. In the end, this allows us to construct the deformation complex for morphisms of algebras over an operad and to represent the deformation ∞-groupoid for differential graded Lie algebras.
Daniel Robert-Nicoud and Bruno Vallette
We present a novel approach to the problem of integrating homotopy Lie algebras by representing the Maurer-Cartan space functor with a universal cosimplicial object. This recovers Getzler's original functor but allows us to prove the existence of additional, previously unknown, structures and properties. Namely, we introduce a well-behaved left adjoint functor, we establish functoriality with respect to infinity-morphisms, and we construct a coherent hierarchy of higher Baker-Campbell-Hausdorff formulas. Thanks to these tools, we are able to establish the most important results of higher Lie theory. We use the recent developments of the operadic calculus, which leads us to explicit tree-wise formulas at all stages; as a consequence we solve the long-standing problem of making explicit a cosimplicial (homotopy) Lie algebra which is Koszul or Eckmann-Hilton dual to the Sullivan algebra of polynomial differential forms on the geometrical simplices. We conclude by applying this theory to rational homotopy theory: the left adjoint functor is shown to provide us with homotopy Lie algebra models which faithfully capture the rational homotopy type of not necessarily connected neither simply-connected topological spaces.
Ricardo Campos, Dan Petersen, Daniel Robert-Nicoud and Felix Wierstra
Over a field of characteristic zero, we show that two commutative differential graded (dg) algebras are quasi-isomorphic if and only if they are quasi-isomorphic as associative dg algebras. This answers a folklore problem in rational homotopy theory, showing that the rational homotopy type of a space is determined by its associative dg algebra of rational cochains. We also show a Koszul dual statement, under an additional completeness hypothesis: two homotopy complete dg Lie algebras whose universal enveloping algebras are quasi-isomorphic as associative dg algebras must themselves be quasi-isomorphic. The latter result applies in particular to nilpotent Lie algebras (not differential graded), in which case it says that two nilpotent Lie algebras whose universal enveloping algebras are isomorphic as associative algebras must be isomorphic.
PhD thesis, 2018
This thesis is divided into two parts. The first one is composed of recollections on operad theory, model categories, simplicial homotopy theory, rational homotopy theory, Maurer-Cartan spaces, and deformation theory. The second part deals with the theory of convolution algebras and some of their applications, as explained below. Suppose we are given a type of algebras, a type of coalgebras, and a relationship between those types of algebraic structures (encoded by an operad, a cooperad, and a twisting morphism respectively). Then, it is possible to endow the space of linear maps from a coalgebra C and an algebra A with a natural structure of Lie algebra up to homotopy. We call the resulting homotopy Lie algebra the convolution algebra of A and C. We study the theory of convolution algebras and their compatibility with the tools of homotopical algebra: infinity morphisms and the homotopy transfer theorem. After doing that, we apply this theory to various domains, such as derived deformation theory and rational homotopy theory. In the first case, we use the tools we developed to construct an universal Lie algebra representing the space of Maurer-Cartan elements, a fundamental object of deformation theory. In the second case, we generalize a result of Berglund on rational models for mapping spaces between pointed topological spaces. In the last chapter of this thesis, we give a new approach to two important theorems in deformation theory: the Goldman-Millson theorem and the Dolgushev-Rogers theorem.
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