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Non functorial cylinders in a model category

100%

García-Calcines J., García-Díaz P., Rodríguez-Machín S.

Open Mathematics

2006

tom 4

nr 3

376-394

Taking cylinder objects, as defined in a model category, we consider a cylinder construction in a cofibration category, which provides a reformulation of relative homotopy in the sense of Baues. Although this cylinder is not a functor we show that it verifies a list of properties which are very closed to those of an I-category (or category with a natural cylinder functor). Considering these new properties, we also give an alternative description of Baues’ relative homotopy groupoids.

Category with a natural cone

84%

Díaz F., Rodríguez-Machín S.

Open Mathematics

2006

tom 4

nr 1

5-33

Generally, in homotopy theory a cylinder object (or, its dual, a path object) is used to define homotopy between morphisms, and a cone object is used to build exact sequences of homotopy groups. Here, an axiomatic theory based on a cone functor is given. Suspension objects are associated to based objects and cofibrations, obtaining homotopy groups referred to an object and relative to a cofibration, respectively. Exact sequences of these groups are built. Algebraic and particular examples are given. We point out that the main results of this paper were already stated in [3], and the purpose of this article is to give full details of the foregoing.

Ograniczanie wyników

2 Open Mathematics

2 Rodríguez-Machín S.

1 Díaz F.

1 García-Calcines J.

1 García-Díaz P.

2 2006

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Non functorial cylinders in a model category

Category with a natural cone