Left-Induced Model Structures and Diagram Categories

Bayeh, Marzieh; Hess, Kathryn; Karpova, Varvara; Kedziorek, Magdalena; Riehl, Emily; Shipley, Brooke

doi:10.1090/conm/641/12859

Cited by 26 publications

(45 citation statements)

References 32 publications

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“…Most recently Drummond-Cole and Hirsh applied the main theorem of [4] to establish the existence of a poset of model category structures on categories of coalgebras over coaugmented, weight-graded cooperads in the category of chain complexes, either unbounded or bounded below, over a field, where the weak equivalences and cofibrations are created by the cobar construction associated to a twisting morphism [12]. Their work generalizes results of Vallette for Koszul cooperads [43].…”

Section: Coalgebras Over Cooperadsmentioning

confidence: 99%

“…Let H be a bimonoid in Ch R , and consider Alg H R , the category of H-comodules in the category of Alg R of differential graded R-algebras. In [4,Theorem 3.8] it was shown that if R is a field, and H is of finite type and non-negatively graded, then the category of non-negatively graded H-comodule algebras (Alg + R ) H admits a model category structure left-induced from the model category structure on Alg + R that is right-induced from the projective structure on Ch R . Here we generalize this result to any commutative ring and any bimonoid H, at the price of working with chain homotopy equivalences rather than quasi-isomorphisms as our weak equivalences.…”

Section: Comodule Algebrasmentioning

confidence: 99%

“…The 'acyclicity condition,' guaranteeing the compatibility of the lifted cofibrations and weak equivalences with fibrations they determine, is formally dual, but the set-theoretic issues are much more complicated. A recent breakthrough result of Makkai and Rosický [30], applied in this context in [4], describes how the set-theoretic obstacles can be overcome. As formulated in [4], a common hypothesis handles the set-theoretic issues in both of the situations above and guarantees that the necessary factorizations can always be constructed.…”

Section: Introductionmentioning

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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Section: Coalgebras Over Cooperadsmentioning

confidence: 99%

Section: Comodule Algebrasmentioning

confidence: 99%

Section: Introductionmentioning

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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“…. , [3], and let sSet 3 = [∆ op 3 , Set]. Left Kan extension, restriction and right Kan extension along the inclusion ∆ 3 ⊆ ∆ gives a chain of adjoints sk 3 tr 3 cosk 3 : sSet 3 → sSet with both sk 3 and cosk 3 fully faithful.…”

Section: Examplesmentioning

confidence: 99%

Bousfield Localisation and Colocalisation of One-Dimensional Model Structures

Balchin¹,

Garner²

2018

Appl Categor Struct

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We give an account of Bousfield localisation and colocalisation for one-dimensional model categories-ones enriched over the model category of 0-types. A distinguishing feature of our treatment is that it builds localisations and colocalisations using only the constructions of projective and injective transfer of model structures along right and left adjoint functors, and without any reference to Smith's theorem.

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“…Earlier work on special cases of model structures on categories of coalgebras can be found in [14], [2], and [13], among others.…”

mentioning

confidence: 99%

The homotopy theory of coalgebras over simplicial comonads

Hess

Kedziorek

2019

Homology, Homotopy and Applications

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We apply the Acyclicity Theorem of [13] (recently corrected in [10]) to establishing the existence of model category structure on categories of coalgebras over comonads arising from simplicial adjunctions, under mild conditions on the adjunction and the associated comonad. We study three concrete examples of such adjunctions where the left adjoint is comonadic and show that in each case the component of the derived counit of the comparison adjunction at any fibrant object is an isomorphism, while the component of the derived unit at any 1-connected object is a weak equivalence. To prove this last result, we explain how to construct explicit fibrant replacements for 1-connected coalgebras in the image of the canonical comparison functor from the Postnikov decompositions of their underlying simplicial sets. We also show in one case that the derived unit is precisely the Bousfield-Kan completion map.

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Left-Induced Model Structures and Diagram Categories

Cited by 26 publications

References 32 publications

A necessary and sufficient condition for induced model structures

A necessary and sufficient condition for induced model structures

Bousfield Localisation and Colocalisation of One-Dimensional Model Structures

The homotopy theory of coalgebras over simplicial comonads

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