Der Hurewicz-Satz

Bauer, Friedrich-Wilhelm

doi:10.2140/pjm.1967.21.1

Cited by 2 publications

(5 citation statements)

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“…Denote the 9K-homotopy class of fie X(X, Y) by [/fe and the set of all Mhomotopy classes in X(X, Y) by [A", Ffe. From 2.2 it follows in the usual way that (1) there is a category JT(SOc) whose objects are those of X and whose morphisms are the 90c-homotopy classes, and (2) the function p : X ->■ XÇSR) which is the identity on objects and sends each/to [f]m, is a covariant functor(4).…”

Section: 2 Propositionmentioning

confidence: 99%

“…The homotopy in F determined by the class 9Jc(<5) is called O-homotopy. The category JT/90c(<I>) is written F* and is essentially the category first studied by Bauer in [1]. Observe that, because of C2, the functor <t> has a unique factorization <D = A o 77 through Jf*.…”

Section: Corollarymentioning

confidence: 99%

“…[Universality] : If F: J¡T -> JSf is any covariant functor to any category if such that Ti<¿) is invertible for each a e W, then there exists a unique covariant functor A: JT/m -+ £ such that F=A o v. 1.1. Theorem.…”

mentioning

confidence: 99%

“…However, no detailed construction of(Jf/Wl, -n) having the generality we require appears in the literature, so we will give a proof of 1.1 in the Appendix. It follows directly from our construction, and we will need this in the sequel, that 1…”

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Categorical Homotopy and Fibrations

Bauer¹,

Dugundji²

1969

Transactions of the American Mathematical Society

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Let X be an arbitrary category, and let 90c be any family of its morphisms. It is known [3] that there is a category X/W (the Gabriel-Zisman "category of fractions of X by 30c ") having the same objects as X, and a covariant functor 7j:Jf-> X/'OJl which is the identity on objects, such that r¡(f) is invertible in X/Wl for each/e 9JÎ.We will use each class 9JÍ to determine a notion of homotopy in X, by defining two morphisms/, g to be 9JÎ-homotopic if -n(f) = rj(g). This notion has the usual properties expected of a homotopy notion; moreover, in the category Top of topological spaces and continuous maps, suitable choices of 9Ji (for example, TO = all homotopy equivalences) reveal 9Jc-homotopy equivalent to the usual notion of homotopy.Each class 501 determines a notion of fibration in X In the category Top, a suitable choice of 9Jt determines both the usual homotopy and the Hurewicz fibrations; however, different classes 9JI may yield the usual notion of homotopy and distinct notions of fibration. More generally, in an arbitrary category X, it is the given class 9JÎ itself, rather than the homotopy notion induced by 9JÏ, that determines the concept of fibration; from this viewpoint, it turns out, surprisingly, that the notion of a Hurewicz fibration is not a homotopy notion. By "reversing arrows," 9JI determines also a concept of cofibration; and again there is a splitting: in fact, two classes may determine the same notion of homotopy but distinct notions of cofibration.In the last section, we introduce the concept of a weak 9Jl-fibration. This notion does not, in general, possess all the advantages of the previous one; however, it reduces to the previous notion under suitable restrictions on the class 9JÎ. Moreover, there are classes, 9JÎ, 9? in the category Top such that {weak W-fibrations} = {Hurewicz fibrations} and {weak 9c-fibrations} = {Dold fibrations}.Each covariant functor O: X -> if determines a O-homotopy in X, by choosing 9J?={/|► tlz determines the usual notion of homotopy;

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Section: 2 Propositionmentioning

confidence: 99%

Section: Corollarymentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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Categorical Homotopy and Fibrations

Bauer¹,

Dugundji²

1969

Transactions of the American Mathematical Society

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show abstract

“…The category JT/90c(<I>) is written F* and is essentially the category first studied by Bauer in [1]. Observe that, because of C2, the functor <t> has a unique factorization <D = A o 77 through Jf*.…”

Section: Corollarymentioning

confidence: 99%