2019
DOI: 10.1007/s00209-019-02305-w
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Equipping weak equivalences with algebraic structure

Abstract: We investigate the extent to which the weak equivalences in a model category can be equipped with algebraic structure. We prove, for instance, that there exists a monad T such that a morphism of topological spaces admits T -algebra structure if and only it is a weak homotopy equivalence. Likewise for quasi-isomorphisms and many other examples. The basic trick is to consider injectivity in arrow categories. Using algebraic injectivity and cone injectivity we obtain general results about the extent to which the … Show more

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Cited by 14 publications
(22 citation statements)
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References 33 publications
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“…More generally, consider a finitary signature Ω : N → Set and the associated function The following is a standard result. The case dealing with a set, rather than a family, of morphisms is dealt with in the proof of Theorem 5 of [8] and the generalisation to a family of morphisms is trivial. Proposition 3.4.…”
Section: Iterated Algebraic Injectives and Models Of Cellular Theoriesmentioning
confidence: 99%
See 1 more Smart Citation
“…More generally, consider a finitary signature Ω : N → Set and the associated function The following is a standard result. The case dealing with a set, rather than a family, of morphisms is dealt with in the proof of Theorem 5 of [8] and the generalisation to a family of morphisms is trivial. Proposition 3.4.…”
Section: Iterated Algebraic Injectives and Models Of Cellular Theoriesmentioning
confidence: 99%
“…Morphisms of the category Sink(Y ) of sinks commute with the coprojections from the Y j in the evident manner. Given a functor 4 In fact, if C is combinatorial, neither condition is required -see Theorem 15 of [8]. However for our present purposes we make use of these conditions and follow Nikolaus original arguments closely.…”
Section: Free Iterated Algebraic Injectives and Cellularitymentioning
confidence: 99%
“…The idea of Corollary 7.4 comes from a passing remark due to Jeff Smith and mentioned in [5]. [4,Theorem 14] contains interesting results of the same kind for more general combinatorial model categories.…”
Section: Reminder About Model Categories Of Fibrant Objectsmentioning
confidence: 99%
“…(X, Y : U ) ( j : X → Y) ( f : X → D) ∃(g : Y → D), g • j = f , so that we get an unspecified extension g of f along j. The algebraic injectivity of D is defined by a given section (−) | j of the restriction map (−) • j, following Bourke's terminology (Bourke, 2017). By − -distributivity, this amounts to (X, Y : U ) ( j :…”
Section: F -mentioning
confidence: 99%