nLab
shape fibration

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A shape fibration is an adaptation of the notion of Hurewicz fibration from the setup of homotopy theory to shape theory (of locally compact metrisable spaces).

A map between metric compacta $p:E\to B$ is called a shape fibration, provided it is induced by a level map $p:E\to B$ of ANR-sequences satisfying the approximate homotopy lifting property with respect to any metric space $X$ and the lifting index, and lifting mesh do not depend on $X$.

If, more generally, $E$ and $B$ are locally compact metric spaces. A proper map $p:E\to B$ is a shape fibration if for any compact $C\subset B$ the restriction of $p$ to a map $p^{-1}(C)\to C$ is a shape fibration between metric compacta.

If $E$ and $B$ are locally compact metric ANR’s then the notion of shape fibration coincides with a notion of an approximate fibration.

• S. Mardešić, T. B. Rushing, Shape fibrations. I. General Topology Appl. 9 (1978), no. 3, 193–215.

• S. Mardešić, J. Segal, (1982) Shape Theory, North Holland.

• T. B. Rushing, Cell-like maps, approximate fibrations and shape fibrations