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Twisting structures and strongly homotopy morphisms
Algebraic Topology · Submitted 12 Jul 2010
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim When the co-unit from K(g) to P is a quasi-isomorphism, we show that the Kleisli category for K(g) is isomorphic to the category of P-algebras and of their morphisms up to strong homotopy, and we give the classifying morphisms for both strict and homotopy P-algebras.
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The abstract, sentence by sentence
In an application of the notion of twisting structures introduced by Hess and Lack, we define twisted composition products of symmetric sequences of chain complexes that are degreewise projective and finitely generated. Let Q be a cooperad and let BP be the bar construction on the operad P. To each morphism of cooperads g from Q to BP is associated a P-co-ring, K(g), which generalizes the two-sided Koszul and bar constructions. When the co-unit from K(g) to P is a quasi-isomorphism, we show that the Kleisli category for K(g) is isomorphic to the category of P-algebras and of their morphisms up to strong homotopy, and we give the classifying morphisms for both strict and homotopy P-algebras. Parametrized morphisms of (co)associative chain (co)algebras up to strong homotopy are also introduced and studied, and a general existence theorem is proved. In the appendix, we study the particular case of the two-sided Koszul resolution of the associative operad.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In an application of the notion of twisting structures introduced by Hess and Lack, we define twisted composition products of symmetric sequences of chain complexes that are degreewise projective and finitely generated. | contribution 0.96 | 0.92 | 0.06 | asserted 1.00 |
| s1 | Let Q be a cooperad and let BP be the bar construction on the operad P. | background 0.54 | 0.22 | 0.02 | asserted 0.65 |
| s2 | To each morphism of cooperads g from Q to BP is associated a P-co-ring, K(g), which generalizes the two-sided Koszul and bar constructions. | contribution 0.95 | 0.82 | 0.08 | asserted 1.00 |
| s3 | When the co-unit from K(g) to P is a quasi-isomorphism, we show that the Kleisli category for K(g) is isomorphic to the category of P-algebras and of their morphisms up to strong homotopy, and we give the classifying morphisms for both strict and homotopy P-algebras. | result 0.98 | 0.95 | 0.68 | asserted 1.00 |
| s4 | Parametrized morphisms of (co)associative chain (co)algebras up to strong homotopy are also introduced and studied, and a general existence theorem is proved. | contribution 0.98 | 0.92 | 0.08 | asserted 1.00 |
| s5 | In the appendix, we study the particular case of the two-sided Koszul resolution of the associative operad. | contribution 0.98 | 0.89 | 0.03 | asserted 0.91 |
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Original abstract
In an application of the notion of twisting structures introduced by Hess and Lack, we define twisted composition products of symmetric sequences of chain complexes that are degreewise projective and finitely generated. Let Q be a cooperad and let BP be the bar construction on the operad P. To each morphism of cooperads g from Q to BP is associated a P-co-ring, K(g), which generalizes the two-sided Koszul and bar constructions. When the co-unit from K(g) to P is a quasi-isomorphism, we show that the Kleisli category for K(g) is isomorphic to the category of P-algebras and of their morphisms up to strong homotopy, and we give the classifying morphisms for both strict and homotopy P-algebras. Parametrized morphisms of (co)associative chain (co)algebras up to strong homotopy are also introduced and studied, and a general existence theorem is proved. In the appendix, we study the particular case of the two-sided Koszul resolution of the associative operad.
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1 arXiv paper cite this in 7 sentences
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Cites it as background 1 paper
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We then remind the reader of the operadic description of $DCSH$, as developed in [ref] and [ref] .
Multiplicative structure in equivariant cohomology · The category $𝐃𝐂𝐒𝐇$ · 0.85
What it cites · 2 sentences without a confident label
Cites as background 2 papers
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Recall [ref] , [ref] that a weight-graded operad is called Koszul if the canonical classifying morphism $κ_P:P^ →P$ is a quasi-isomorphism.
Operads, homotopy algebra and iterated integrals for double loop spaces · Weight-graded operads · 0.98
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Earlier versions of our approach to describing strongly homotopy morphisms via co-rings can be found on the arXiv [ref] .
math.AT/0505559 · Acknowledgments · 0.86
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