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Is the shape theory of Hausdorff compact spaces complete with respect to the inverse limit operation?--complete means that for every inverse system of Hausdorff compact spaces, and the shape morphisms between them, there exists a Hausdorff compact space which is the (inverse) limit (in the shape category) of the system. Remember that the shape morphisms do not have to be--and in general they are not--induced by continuous maps.

WARNING: the shape theory of Hausdorff compact spaces is continuous with respect to inverse limit; this (by itself) doesn't mean that the shape theory of Hausdorff compact spaces is complete--as long as I know, it is as open question as it was since the time when the shape theory of Hausdorff compact spaces was defined.

REMARK: the inverse limit of a shape inverse sequence of Hausdorff compact spaces always exists; the countable case is fine.

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    $\begingroup$ What does 'complete' mean here? $\endgroup$
    – Fernando Muro
    Commented May 21, 2015 at 7:51
  • $\begingroup$ @FernandoMuro -- I have expanded the text of my question to answer yours. $\endgroup$
    – Włodzimierz Holsztyński
    Commented May 21, 2015 at 9:21
  • $\begingroup$ You're then talkin about the completeness of a certain category rathen than theory. $\endgroup$
    – Fernando Muro
    Commented May 21, 2015 at 9:45
  • $\begingroup$ @FernandoMuro -- right, it'd be more precise. However shape category and shape theory are sometimes used as synonyms (shape theory rolls the tongue smoother, I think). $\endgroup$
    – Włodzimierz Holsztyński
    Commented May 21, 2015 at 10:06
  • $\begingroup$ As the wiki entry to shape theory is not very expansive, could you state your question in "layman's terms"? $\endgroup$
    – Dominic van der Zypen
    Commented May 21, 2015 at 14:49

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