Lie models of simplicial sets and representability of the Quillen functor

Buijs, Urtzi; Félix, Yves; Murillo, Aniceto; Tanré, Daniel

doi:10.1007/s11856-020-2026-8

Cited by 8 publications

(26 citation statements)

References 27 publications

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“…In we proved: Theorem There is a unique (up to isomorphism) cdgl of the form,

{frakturL}_{Δ^{n}} = (\overset{L}{true} ((), s^{- 1}, C_{*}, false(Δ^{n} false)), d),

such that (1)for each

i = 0, \dots, n

, the generators

a_{i} \in s^{- 1} C_{0} false(Δ^{n} false)

, corresponding to vertices, are Maurer–Cartan elements; (2)the linear part

d_{1}

d

is precisely the desuspension of

δ

; (3)the maps induced by the cofaces are the previous

σ_{j} : {frakturL}_{Δ^{n}} \to {frakturL}_{Δ^{n - 1}}

. …”

Section: The Model and Realization Functorsmentioning

confidence: 99%

“…Remark (i)The inverse isomorphism

ψ^{- 1} : false(\overset{L}{true} (a, u, s u), d false) \overset{≅}{⟶} false(\overset{L}{true} (a, b, x), d false)

is given by

ψ^{- 1} false(a false) = a

ψ^{- 1} false(s u false) = x

and

ψ^{- 1} false(u false) = \partial x

. (ii)The differential of

L_{{normalΔ}^{1}}

is entirely determined by the differentials of

a

and

b

, and by the linear part of the differential of

x

, see [, Theorem 1.4; , Main Theorem]. …”

Section: The Model and Realization Functorsmentioning

confidence: 99%

“…On the other hand

{frakturL}_{Δ^{•}} = {{frakturL}_{Δ^{n}}}_{n ⩾ 0}

has a cosimplicial cdgl structure [, § 3] which gives rise to the following. Definition The realization of a cdgl

L

is defined as the simplicial set

false⟨ L false⟩ = {Hom}_{cDGL} false(L_{{normalΔ}^{•}}, L false) .

…”

Section: The Model and Realization Functorsmentioning

confidence: 99%

“…In , we have built a pair of adjoint functors, model and realization ,

between the usual category of simplicial sets and the category of complete differential graded Lie algebras (for short cdgl algebras) defined over

double-struckQ

. This category and the main properties of these functors are described in detail in Sections 1 and 2, respectively.…”

Section: Introductionmentioning

confidence: 99%

“…When

X

is a finite simply connected simplicial set, the component

L_{X}^{a}

of the model

L_{X}

at a non‐trivial Maurer–Cartan element

a

(see Definition ) is quasi‐isomorphic to the Quillen rational model of

X

[, Theorem C]. In particular, in view of equality and Theorems and , we deduce that

false⟨ L_{X} false⟩ ≃ X_{double-struckQ}^{+}

, the disjoint union of the rationalization of

X

with an external point.…”

Section: Introductionmentioning

confidence: 99%

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Homotopy theory of complete Lie algebras and Lie models of simplicial sets

Buijs

Félix

Murillo

et al. 2018

Journal of Topology

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In a previous work, by extending the classical Quillen construction to the non-simply connected case, we have built a pair of adjoint functors, model and realization,cDGL between the categories of simplicial sets and complete differential graded Lie algebras. This paper is a follow-up of this work. We show that when X is a finite connected simplicial set, then LX coincides with Q∞X + , the disjoint union of the Bousfield-Kan completion of X with an external point. We also define a model category structure on cDGL making L and − a Quillen pair, and we construct an explicit cylinder. In particular, these functors preserve homotopies and weak equivalences and therefore, this gives the basis for developing a cDGL rational homotopy theory for all spaces. Contents

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“…In we proved: Theorem There is a unique (up to isomorphism) cdgl of the form,

{frakturL}_{Δ^{n}} = (\overset{L}{true} ((), s^{- 1}, C_{*}, false(Δ^{n} false)), d),

such that (1)for each

i = 0, \dots, n

, the generators

a_{i} \in s^{- 1} C_{0} false(Δ^{n} false)

, corresponding to vertices, are Maurer–Cartan elements; (2)the linear part

d_{1}

d

is precisely the desuspension of

δ

; (3)the maps induced by the cofaces are the previous

σ_{j} : {frakturL}_{Δ^{n}} \to {frakturL}_{Δ^{n - 1}}

. …”

Section: The Model and Realization Functorsmentioning

confidence: 99%

“…Remark (i)The inverse isomorphism

ψ^{- 1} : false(\overset{L}{true} (a, u, s u), d false) \overset{≅}{⟶} false(\overset{L}{true} (a, b, x), d false)

is given by

ψ^{- 1} false(a false) = a

ψ^{- 1} false(s u false) = x

and

ψ^{- 1} false(u false) = \partial x

. (ii)The differential of

L_{{normalΔ}^{1}}

is entirely determined by the differentials of

a

and

b

, and by the linear part of the differential of

x

, see [, Theorem 1.4; , Main Theorem]. …”

Section: The Model and Realization Functorsmentioning

confidence: 99%

“…On the other hand

{frakturL}_{Δ^{•}} = {{frakturL}_{Δ^{n}}}_{n ⩾ 0}

has a cosimplicial cdgl structure [, § 3] which gives rise to the following. Definition The realization of a cdgl

L

is defined as the simplicial set

false⟨ L false⟩ = {Hom}_{cDGL} false(L_{{normalΔ}^{•}}, L false) .

…”

Section: The Model and Realization Functorsmentioning

confidence: 99%

“…In , we have built a pair of adjoint functors, model and realization ,

between the usual category of simplicial sets and the category of complete differential graded Lie algebras (for short cdgl algebras) defined over

double-struckQ

. This category and the main properties of these functors are described in detail in Sections 1 and 2, respectively.…”

Section: Introductionmentioning

confidence: 99%

“…When

X

is a finite simply connected simplicial set, the component

L_{X}^{a}

of the model

L_{X}

at a non‐trivial Maurer–Cartan element

a

(see Definition ) is quasi‐isomorphic to the Quillen rational model of

X

[, Theorem C]. In particular, in view of equality and Theorems and , we deduce that

false⟨ L_{X} false⟩ ≃ X_{double-struckQ}^{+}

, the disjoint union of the rationalization of

X

with an external point.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Homotopy theory of complete Lie algebras and Lie models of simplicial sets

Buijs

Félix

Murillo

et al. 2018

Journal of Topology

Self Cite

View full text Add to dashboard Cite

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Lie models for nilpotent spaces

2018

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Let (L, d) be a differential graded Lie algebra, where L = L(V ) is free as graded Lie algebra and V = V ≥0 is a finite type graded vector space. We prove that the injection of (L, d) into its completion ( L, d) is a quasi-isomorphism if and only if H (L, d) is a finite type pronilpotent graded Lie algebra. As a consequence, we obtain an equivalence between graded Lie models for nilpotent spaces in rational homotopy theory.

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