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Do you have a reference for or explanation of that? There are a few points which are not obvious to me:
a) I have never heard of a dual to the Hurewicz/Strøm model structure.
b) It is not obvious to me that there is any 2-monad around, how is it defined exactly?
c) Some non-trivial argument must be needed to show that the iso-fibrations are Hurewicz fibrations in the usual sense.
That the Hurewicz model structure fits into the same circle of ideas is clear (in particular, one can view Hurewicz fibrations as algebras for an endofunctor defined using Moore paths, as in Barthel and Riehl’s work), but my impression has always been that there is a fundamental lack of duality in the details of the construction of the Strøm model structure, which prevents it from fitting exactly into these kind of frameworks. E.g. most of my thesis applies to the Strøm model structure, but not everything, because I have found no way to cook up the necessary strictness in a suitably dualisable way: Moore paths do not dualise very well (there is no obvious way to define a ’Moore cylinder’, although for finite/Alexandroff topological spaces one can maybe do something).
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