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1. • CommentRowNumber1.
   • CommentAuthorTim_Porter
   • CommentTimeSep 11th 2012
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexThere was an interesting post on the café about ultra filters and category theory. I remember years ago working on the use of equational compactness and its relationship with the vanishing of the derived functors of the inverse limit functor. There was a result something like: if you take Cech homology on compact metric spaces with coefficients in some module M, then the homology 'exact sequence' is always exact exactly when M is equationally compact. (Garavaglia, Fund. Math. 100(1978)89-95). There was a SLN on equational compactness for rings (No 745) [David K Haley- 1976, but it is a long time since I read it. There was also a connection with results in proper homotopy theory and strong shape theory where Ed Brown's proper homotopy groups were related to a construction using ultra powers. Both these sorts of results relate to homotopy limits, so that increases the likelihood that there is some as yet hidden connection between infinity category theory and this ultrafilter stuff. I thought it worth while posting this here to see if anyone else knows further material of relevance.

   There was an interesting post on the café about ultra filters and category theory. I remember years ago working on the use of equational compactness and its relationship with the vanishing of the derived functors of the inverse limit functor. There was a result something like: if you take Cech homology on compact metric spaces with coefficients in some module M, then the homology ’exact sequence’ is always exact exactly when M is equationally compact. (Garavaglia, Fund. Math. 100(1978)89-95). There was a SLN on equational compactness for rings (No 745) [David K Haley- 1976, but it is a long time since I read it.

   There was also a connection with results in proper homotopy theory and strong shape theory where Ed Brown’s proper homotopy groups were related to a construction using ultra powers. Both these sorts of results relate to homotopy limits, so that increases the likelihood that there is some as yet hidden connection between infinity category theory and this ultrafilter stuff.

   I thought it worth while posting this here to see if anyone else knows further material of relevance.