Abstract Homotopy and Simple Homotopy Theory

Kamps, Klaus; Porter, Timothy

doi:10.1142/2215

Cited by 67 publications

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“…Our weak fibrations, and their properties, depend on this cylinder object choice. In case this choice comes from a cylinder functor satisfying certain Kan filler conditions, our fibre homotopy relation coincides with the one used in [KP97]. This makes it possible to compare our weak fibrations with Kamps's.…”

Section: Introductionmentioning

confidence: 92%

“…Booth [Boo93] also obtains versions of Dold's theorem, using suitably defined generalizations of the covering homotopy property. In other cases the fibre homotopy equivalences were studied in a categorical setting, as for example in the paper [HKK96] by HardieKamps-Kieboom and the book [KP97] of Kamps-Porter. Homotopy structure can be imposed on an appropriate category in several ways.…”

Section: Introductionmentioning

confidence: 99%

“…Homotopy structure can be imposed on an appropriate category in several ways. In [HKK96] and [KP97] the basic assumption is that the category has some cylinder functor. In the article [Kam72], Kamps uses cylinder functors to define a notion of weak fibration.…”

Section: Introductionmentioning

confidence: 99%

“…These two methods feature in categories of cofibrant objects and, respectively, categories of fibrant objects. Of these two notions, the latter was introduced by K. S. Brown [Bro73] in 1973 and dualized into the former by Kamps and Porter (see [KP97]). We consider a notion of weak fibration in the context of a category of cofibrant objects with a cylinder object choice, i.e., a chosen cylinder object for every object of the category.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 1 we recall the axioms of a category of cofibrant objects and introduce the notion of cylinder object choice. The definition of fibre homotopy from [KP97] is adapted to our context. Based on one of the equivalent formulations-due to Kieboom [Kie87]-of the concept of weak fibration in the topological case, for a category of cofibrant objects we define the concept of weak fibration in terms of the so-called weak right lifting property.…”

Section: Introductionmentioning

confidence: 99%

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Weak (co)fibrations in categories of (co)fibrant objects

Kieboom

Sonck

Linden

et al. 2003

Homology, Homotopy and Applications

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We introduce a fibre homotopy relation for maps in a category of cofibrant objects equipped with a choice of cylinder objects. Weak fibrations are defined to be those morphisms having the weak right lifting property with respect to weak equivalences. We prove a version of Dold's fibre homotopy equivalence theorem and give a number of examples of weak fibrations. If the category of cofibrant objects comes from a model category, we compare fibrations and weak fibrations, and we compare our fibre homotopy relation, which is defined in terms of left homotopies and cylinders, with the fibre homotopy relation defined in terms of right homotopies and path objects. We also dualize our notion of weak fibration in a category of cofibrant objects to a notion of weak cofibration in a category of fibrant objects, and give examples of these weak cofibrations. A section is devoted to the case of chain complexes in an abelian category. IntroductionThe fibre homotopy equivalence theorem of Dold [Dol63, Theorem 6.1] in Top has been generalized by various authors. Besides the original work by Dold, the book [DKP70] of tom Dieck-Kamps-Puppe gives an exposition on weak fibrations (h-Faserungen in Top). Some of the generalizations consider maps which are simultaneously over a given space and under a given space. Booth [Boo93] also obtains versions of Dold's theorem, using suitably defined generalizations of the covering homotopy property. In other cases the fibre homotopy equivalences were studied in a categorical setting, as for example in the Homology, Homotopy and Applications, vol. 5(1), 2003 346 the basic assumption is that the category has some cylinder functor. In the article [Kam72], Kamps uses cylinder functors to define a notion of weak fibration. A model category structure, a concept due to Quillen [Qui67], is another way of introducing a homotopy relation in a category. In fact in a model category there are two dual ways of defining homotopy of maps: left homotopies, defined in terms of cylinder objects, and right homotopies, defined in terms of cocylinder objects. These two methods feature in categories of cofibrant objects and, respectively, categories of fibrant objects. Of these two notions, the latter was introduced by K. S. Brown [Bro73] in 1973 and dualized into the former by Kamps and Porter (see [KP97]). We consider a notion of weak fibration in the context of a category of cofibrant objects with a cylinder object choice, i.e., a chosen cylinder object for every object of the category. Our weak fibrations, and their properties, depend on this cylinder object choice. In case this choice comes from a cylinder functor satisfying certain Kan filler conditions, our fibre homotopy relation coincides with the one used in [KP97]. This makes it possible to compare our weak fibrations with Kamps's.The aim of this article is to study fibre homotopies and weak fibrations in a category of cofibrant objects and, dually, relative homotopies and weak cofibrations in a category of fibrant objects. The presentation is a...

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Section: Introductionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%