Cooperads and coalgebras as closed model categories

Aubry, Marc; Chataur, David

doi:10.1016/s0022-4049(02)00174-3

Cited by 12 publications

(17 citation statements)

References 9 publications

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“…In 2003, Aubry and Chataur proved the existence of model category structures on (certain) cooperads and coalgebras over them in unbounded chain complexes over a field [3]. Smith established results along the same lines in [24] in 2011.…”

Section: Related Workmentioning

confidence: 93%

The homotopy theory of coalgebras over a comonad

Hess

Shipley²

2013

Proceedings of the London Mathematical Society

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Abstract. Let K be a comonad on a model category M. We provide conditions under which the associated category M K of K-coalgebras admits a model category structure such that the forgetful functor M K → M creates both cofibrations and weak equivalences.We provide concrete examples that satisfy our conditions and are relevant in descent theory and in the theory of Hopf-Galois extensions. These examples are specific instances of the following categories of comodules over a coring (coring). For any semihereditary commutative ring R, let A be a dg R-algebra that is homologically simply connected. Let V be an A-coring that is semifree as a left A-module on a degreewise R-free, homologically simply connected graded module of finite type. We show that there is a model category structure on the category M A of right A-modules satisfying the conditions of our existence theorem with respect to the comonad − ⊗ A V and conclude that the category M V A of V -comodules in M A admits a model category structure of the desired type. Finally, under extra conditions on R, A, and V , we describe fibrant replacements in M V A in terms of a generalized cobar construction.

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Section: Related Workmentioning

confidence: 93%

The homotopy theory of coalgebras over a comonad

Hess

Shipley²

2013

Proceedings of the London Mathematical Society

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“…The listed model category structure on dggc is implied by general operad theory work of [1,Thm 3.2.3]. That fibrations are indeed the cofree maps follows in the finitely generated case from the dual of the corresponding statement about cofibrations in dgla.…”

Section: ])mentioning

confidence: 98%

“…As in the dgcc and dgla settings, the structures given in Theorem B.5 (minus the description of fibrations in dggc) give closed model category structures when finiteness assumptions are removed. There is a discrepancy between this situation and that of [1], which defines cooperads using direct sums and orbits instead of products and fixed points.…”

Section: ])mentioning

confidence: 99%

Lie coalgebras and rational homotopy theory, I: graph coalgebras

Sinha

Walter

2011

Homology, Homotopy and Applications

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We give a new presentation of the Lie cooperad as a quotient of the graph cooperad, a presentation which is not linearly dual to any of the standard presentations of the Lie operad. We use this presentation to explicitly compute duality between Lie algebras and coalgebras, to give a new presentation of Harrison homology, and to show that Lyndon words yield a canonical basis for cofree Lie coalgebras.

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“…One of the motivations for writing this note was the paper [3]. In [4], the authors extended the main result of [3] to the category of cooperads, or F 2 -comonoids, in the category of non-negatively graded chain complexes of vector spaces. We do not know whether the technique used in this paper would provide a model structure on the category of cooperads in E .…”

Section: Which Is a Weak Equivalence If Either One Of I Or I Is;mentioning

confidence: 99%