Cellular categories

Makkai, Michael; Rosický, Jiřı́

doi:10.1016/j.jpaa.2014.01.005

Cited by 27 publications

(46 citation statements)

References 18 publications

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“…The proof of part (1) depends crucially on the fact that accessible categories and accessible functors are closed under a certain class of 2-categorical limit constructions. This is closely related to the essential ingredient in Theorem 2.23 of [3], which is a result of Makkai and Rosicky [10] that states that locally presentable categories with a cofibrantly generated weak factorization system, and appropriate functors between such, are closed under the same class of 2-categorical limits. We defer this part of our proof to the appendix, allowing for a more leisurely treatment.…”

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confidence: 73%

“…The main theorem of Makkai and Rosický [10] is that the 2-category of locally presentable categories equipped with a cofibrantly generated weak factorization system, colimit-preserving functors that preserve the left classes, and all natural transformations has PIE-limits created by the forgetful functor to CAT. The pseudopullback of a cofibrantly generated weak factorization system on C along a colimit-preserving functor u : A → C between locally presentable categories equipped with 'trivial' weak factorization systems defines the left-induced weak factorization system on A, which is therefore cofibrantly generated.…”

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confidence: 99%

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Coalgebraic models for combinatorial model categories

Ching

Riehl

2014

Homology, Homotopy and Applications

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We show that the category of algebraically cofibrant objects in a combinatorial and simplicial model category A has a model structure that is left-induced from that on A. In particular it follows that any presentable model category is Quillen equivalent (via a single Quillen equivalence) to one in which all objects are cofibrant. such λ > µ ′ , conditions (1) and (2) of the lemma are satisfied.Suppose that X ∈ A is λ-presentable and (Y j ) j∈J is a λ-filtered diagram in A. By construction of λ, X ⊗ ∆ n is also λ-presentable and so

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confidence: 73%

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confidence: 99%

Coalgebraic models for combinatorial model categories

Ching

Riehl

2014

Homology, Homotopy and Applications

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“…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%

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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“…Proposition 5 below) that mixing of model structures may be reduced in turn to liftings, so that we have a construction of localisations from projective and injective liftings alone. Note that this approach does not avoid the cardinality arguments involved in Smith's theorem; rather, it pushes them elsewhere, namely into the construction of injective liftings of model structures as detailed in [22]. In particular, our approach gives no more of an explicit grasp on the classes of maps of a localisation than the usual one.…”

Section: Introductionmentioning

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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Cellular categories

Cited by 27 publications

References 18 publications

Coalgebraic models for combinatorial model categories

Coalgebraic models for combinatorial model categories

A necessary and sufficient condition for induced model structures

Bousfield Localisation and Colocalisation of One-Dimensional Model Structures

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