scholarly journals The Milnor-Moore theorem for $$L_\infty $$ algebras in rational homotopy theory

2021 ◽  
Author(s):  
José Manuel Moreno Fernández
Keyword(s):  
Homotopy Theory ◽  
Homotopy Groups ◽  
Rational Homotopy Theory ◽  
Rational Homotopy ◽  
Massey Products ◽  
Homotopy Types ◽  
Homology Algebra ◽  
Simply Connected ◽  
Space Homology ◽  
Loop Space Homology

AbstractWe give a construction of the universal enveloping $$A_\infty $$ A ∞ algebra of a given $$L_\infty $$ L ∞ algebra, alternative to the already existing versions. As applications, we derive a higher homotopy algebras version of the classical Milnor-Moore theorem. This proposes a new $$A_\infty $$ A ∞ model for simply connected rational homotopy types, and uncovers a relationship between the higher order rational Whitehead products in homotopy groups and the Pontryagin-Massey products in the rational loop space homology algebra.

scholarly journals Goodwillie Approximations to Higher Categories

2021 ◽  
Vol 272 (1333) ◽  
Author(s):  
Gijs Heuts
Keyword(s):  
Homotopy Theory ◽  
Symmetric Groups ◽  
Homotopy Groups ◽  
Rational Homotopy Theory ◽  
Rational Homotopy ◽  
Content Type ◽  
Simply Connected ◽  
Derivatives Of ◽  
Connected Spaces

We construct a Goodwillie tower of categories which interpolates between the category of pointed spaces and the category of spectra. This tower of categories refines the Goodwillie tower of the identity functor in a precise sense. More generally, we construct such a tower for a large class of ∞ \infty -categories C \mathcal {C} and classify such Goodwillie towers in terms of the derivatives of the identity functor of C \mathcal {C} . As a particular application we show how this provides a model for the homotopy theory of simply-connected spaces in terms of coalgebras in spectra with Tate diagonals. Our classification of Goodwillie towers simplifies considerably in settings where the Tate cohomology of the symmetric groups vanishes. As an example we apply our methods to rational homotopy theory. Another application identifies the homotopy theory of p p -local spaces with homotopy groups in a certain finite range with the homotopy theory of certain algebras over Ching’s spectral version of the Lie operad. This is a close analogue of Quillen’s results on rational homotopy.

scholarly journals Rational homotopy theory for non-simply connected spaces

1999 ◽  
Vol 352 (4) ◽  
pp. 1493-1525 ◽  
Author(s):  
Antonio Gómez-Tato ◽  
Stephen Halperin ◽  
Daniel Tanré
Keyword(s):  
Homotopy Theory ◽  
Rational Homotopy Theory ◽  
Rational Homotopy ◽  
Simply Connected ◽  
Connected Spaces

scholarly journals Schematic homotopy types and non-abelian Hodge theory

2008 ◽  
Vol 144 (3) ◽  
pp. 582-632 ◽  
Author(s):  
L. Katzarkov ◽  
T. Pantev ◽  
B. Toën
Keyword(s):  
Homotopy Theory ◽  
Main Tool ◽  
Hodge Decomposition ◽  
Homotopy Groups ◽  
Projective Manifold ◽  
Hodge Structures ◽  
Homotopy Types ◽  
Projective Manifolds ◽  
Simply Connected ◽  
Complex Projective

AbstractWe use Hodge theoretic methods to study homotopy types of complex projective manifolds with arbitrary fundamental groups. The main tool we use is the schematization functor$X \mapsto (X\otimes \mathbb {C})^{\mathrm {sch}}$, introduced by the third author as a substitute for the rationalization functor in homotopy theory in the case of non-simply connected spaces. Our main result is the construction of a Hodge decomposition on $(X\otimes \mathbb {C})^{\mathrm {sch}}$. This Hodge decomposition is encoded in an action of the discrete group $\mathbb {C}^{\times \delta }$ on the object $(X\otimes \mathbb {C})^{\mathrm {sch}}$ and is shown to recover the usual Hodge decomposition on cohomology, the Hodge filtration on the pro-algebraic fundamental group, and, in the simply connected case, the Hodge decomposition on the complexified homotopy groups. We show that our Hodge decomposition satisfies a purity property with respect to a weight filtration, generalizing the fact that the higher homotopy groups of a simply connected projective manifold have natural mixed Hodge structures. As applications we construct new examples of homotopy types which are not realizable as complex projective manifolds and we prove a formality theorem for the schematization of a complex projective manifold.

On equivariant disconnected rational homotopy theory

2010 ◽  
Vol 17 (2) ◽  
pp. 229-240
Author(s):  
Marek Golasiński
Keyword(s):  
Homotopy Theory ◽  
Weak Equivalence ◽  
Rational Homotopy Theory ◽  
Rational Homotopy ◽  
Hamiltonian Group ◽  
Simplicial Set ◽  
De Rham

Abstract An equivariant disconnected Sullivan–de Rham equivalence is developed using Kan's result on diagram categories. Given a finite Hamiltonian group G, let X be a G-simplicial set. It is shown that the associated system of algebras indexed by the category 𝒪(G) of a canonical orbit can be “approximated” (up to a weak equivalence) by such a system ℳ X with the properties required by nonequivariant minimal algebras.

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