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Fibrations in the Category of Absolute Neighborhood Retracts

100%

Miyata T.

Bulletin of the Polish Academy of Sciences. Mathematics

2007

tom 55

nr 2

145-154

The category Top of topological spaces and continuous maps has the structures of a fibration category and a cofibration category in the sense of Baues, where fibration = Hurewicz fibration, cofibration = the usual cofibration, and weak equivalence = homotopy equivalence. Concentrating on fibrations, we consider the problem: given a full subcategory 𝓒 of Top, is the fibration structure of Top restricted to 𝓒 a fibration category? In this paper we take the special case where 𝓒 is the full subcategory ANR of Top whose objects are absolute neighborhood retracts. The main result is that ANR has the structure of a fibration category if fibration = map having a property that is slightly stronger than the usual homotopy lifting property, and weak equivalence = homotopy equivalence.

Dimension-raising maps in a large scale

63%

Miyata T., Virk Ž.

Fundamenta Mathematicae

2013

tom 223

nr 1

83-97

Hurewicz's dimension-raising theorem states that dim Y ≤ dim X + n for every n-to-1 map f: X → Y. In this paper we introduce a new notion of finite-to-one like map in a large scale setting. Using this notion we formulate a dimension-raising type theorem for asymptotic dimension and asymptotic Assouad-Nagata dimension. It is also well-known (Hurewicz's finite-to-one mapping theorem) that dim X ≤ n if and only if there exists an (n+1)-to-1 map from a 0-dimensional space onto X. We formulate a finite-to-one mapping type theorem for asymptotic dimension and asymptotic Assouad-Nagata dimension.

Ograniczanie wyników

1 Bulletin of the Polish Academy of Sciences. Mathematics

1 Fundamenta Mathematicae

2 Miyata T.

1 Virk Ž.

1 2013

1 2007

Wyniki wyszukiwania

Fibrations in the Category of Absolute Neighborhood Retracts

Dimension-raising maps in a large scale