The Hochschild complex of a twisting cochain

Hess, Kathryn

doi:10.1016/j.jalgebra.2015.11.040

Cited by 6 publications

(11 citation statements)

References 24 publications

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“…Proof of Corollary Note that any bialgebra is naturally augmented (by its counit) as an algebra and coaugmented (by its unit) as a coalgebra. By [, Theorem 3.12], the algebra

Ω Bar H

admits a natural bialgebra structure with respect to which

ε_{H}

is a morphism of bialgebras. Dually, the coalgebra

Bar Ω H

admits a natural bialgebra structure with respect to which

η_{H}

is a morphism of bialgebras.…”

Section: Dg Examplesmentioning

confidence: 99%

“…Moreover it is likely that a counter‐example similar to that above can be constructed in this context, establishing that

{false((P, Q) - {Bialg}_{R} false)}_{⌝}

and

{false((P, Q) - {Bialg}_{R} false)}_{⌞}

are indeed distinct. Proving analogues of Corollary and of the counter‐example above would require a generalization of [, Theorem 3.12], that is, that if

H

is a

(P, Q)

‐bialgebra, then

{normalΩ}_{τ} {prefixBar}_{τ} H

and

{prefixBar}_{τ} {normalΩ}_{τ} H

both admit natural

(P, Q)

‐bialgebra structures. We suspect that this is the case, at least under reasonable conditions on the twisting morphism

τ

, but the proof is beyond the scope of this article.…”

Section: Dg Examplesmentioning

confidence: 99%

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A necessary and sufficient condition for induced model structures

Hess

Kedziorek

Riehl

et al. 2017

Journal of Topology

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A common technique for producing a new model category structure is to lift the fibrations and weak equivalences of an existing model structure along a right adjoint. Formally dual but technically much harder is to lift the cofibrations and weak equivalences along a left adjoint. For either technique to define a valid model category, there is a well-known necessary "acyclicity" condition. We show that for a broad class of "accessible model structures" - a generalization introduced here of the well-known combinatorial model structures - this necessary condition is also sufficient in both the right-induced and left-induced contexts, and the resulting model category is again accessible. We develop new and old techniques for proving the acyclity condition and apply these observations to construct several new model structures, in particular on categories of differential graded bialgebras, of differential graded comodule algebras, and of comodules over corings in both the differential graded and the spectral setting. We observe moreover that (generalized) Reedy model category structures can also be understood as model categories of "bialgebras" in the sense considered here.Comment: 49 pages; final journal version to appear in the Journal of Topolog

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“…Proof of Corollary Note that any bialgebra is naturally augmented (by its counit) as an algebra and coaugmented (by its unit) as a coalgebra. By [, Theorem 3.12], the algebra

Ω Bar H

admits a natural bialgebra structure with respect to which

ε_{H}

is a morphism of bialgebras. Dually, the coalgebra

Bar Ω H

admits a natural bialgebra structure with respect to which

η_{H}

is a morphism of bialgebras.…”

Section: Dg Examplesmentioning

confidence: 99%

“…Moreover it is likely that a counter‐example similar to that above can be constructed in this context, establishing that

{false((P, Q) - {Bialg}_{R} false)}_{⌝}

and

{false((P, Q) - {Bialg}_{R} false)}_{⌞}

are indeed distinct. Proving analogues of Corollary and of the counter‐example above would require a generalization of [, Theorem 3.12], that is, that if

H

is a

(P, Q)

‐bialgebra, then

{normalΩ}_{τ} {prefixBar}_{τ} H

and

{prefixBar}_{τ} {normalΩ}_{τ} H

both admit natural

(P, Q)

‐bialgebra structures. We suspect that this is the case, at least under reasonable conditions on the twisting morphism

τ

, but the proof is beyond the scope of this article.…”

Section: Dg Examplesmentioning

confidence: 99%

A necessary and sufficient condition for induced model structures

Hess

Kedziorek

Riehl

et al. 2017

Journal of Topology

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“…The coHochschild complex of

C

, as defined in , is the dg

double-struckk

‐module

Λ C = (C \otimes Ω C, d_{Λ C})

with differential

d_{Λ C} = d_{C} \otimes 1 + 1 \otimes d_{Ω C} + θ_{1} + θ_{2}

where

\begin{matrix} true \\ right θ_{1} (v \otimes [{\bar{c}}_{1} | \dots | {\bar{c}}_{n}]) & = & left - \sum {(- 1)}^{false| v^{'} false|} v^{'} \otimes false[\bar{v^{''}} false| {\overset{c}{true}}_{1} false| \dots false| {\overset{c}{true}}_{n} false], \\ right θ_{2} (v \otimes [{\bar{c}}_{1} | \dots | {\bar{c}}_{n}]) & = & left \sum {(- 1)}^{false(false| v^{'} false| + 1 false) false(false| v^{''} false| + ε_{n}^{c} false)} v^{''} \otimes false[{\overset{c}{true}}_{1} false| \dots false| {\overset{c}{true}}_{n} false| \overset{v^{'}}{true} false], \\ right ε_{n}^{x} & = & left \end{matrix}

…”

Section: Algebraic Models For the Free Loop Space And The Hat‐cohochsmentioning

confidence: 99%

“…The following result was stated in , however, we outline a proof not relying on the comparison theorem for spectral sequences of twisted tensor products, which assumes certain hypotheses. Proposition Let

C

be a connected dgc coalgebra.…”

Section: Algebraic Models For the Free Loop Space And The Hat‐cohochsmentioning

confidence: 99%

“…The above construction is part of the following general fact shown in [6], which we do not need in full generality for our purposes: ρ is actually a morphism of coalgebras up to strong homotopy compatible with twisting cochains, and these kind of morphisms induce chain maps between the more general notion of Hochschild complexes associated to a twisting cochain. There is a 'local' chain contraction s : Kerρ → Kerρ defined for x := [x 1 | .…”

Section: The Hat-cohochschild Constructionmentioning

confidence: 99%

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A combinatorial model for the free loop fibration

Rivera

Saneblidze

2018

Bull. London Math. Soc.

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We introduce the abstract notion of a closed necklical set in order to describe a functorial combinatorial model of the free loop fibration ΩY→ΛY→Y over the geometric realization Y=|X| of a path‐connected simplicial set X. In particular, to any path‐connected simplicial set X we associate a closed necklical set normalΛ̂X such that its geometric realization false|normalΛ̂Xfalse|, a space built out of glueing ‘freehedrical’ and ‘cubical’ cells, is homotopy equivalent to the free loop space ΛY and the differential graded module of chains C∗false(trueΛ̂Xfalse) generalizes the coHochschild chain complex of the chain coalgebra C*false(Xfalse).

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Twisting structures and morphisms up to strong homotopy

Hess

Parent

Scott

2019

J. Homotopy Relat. Struct.

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We define twisted composition products of symmetric sequences via classifying morphisms rather than twisting cochains. Our approach allows us to establish an adjunction that simultaneously generalizes a classic one for algebras and coalgebras, and the barcobar adjunction for quadratic operads. The comonad associated to this adjunction turns out to be, in several cases, a standard Koszul construction. The associated Kleisli categories are the "strong homotopy" morphism categories. In an appendix, we study the co-ring associated to the canonical morphism of cooperads A ⊥ → BA, which is exactly the two-sided Koszul resolution of the associative operad A, also known as the Alexander-Whitney co-ring.

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The Hochschild complex of a twisting cochain

Cited by 6 publications

References 24 publications

A necessary and sufficient condition for induced model structures

A necessary and sufficient condition for induced model structures

A combinatorial model for the free loop fibration

Twisting structures and morphisms up to strong homotopy

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