Defines r-quasi-isomorphisms and r-cofibrations on generalized metric spaces so that each page of the magnitude-path spectral sequence satisfies metric Eilenberg-Steenrod axioms and supports Brown category structures for homotopy colimits, restricting to directed graphs at r=1.
Cofibrations in Homotopy Theory
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abstract
We define Anderson-Brown-Cisinski (ABC) cofibration categories, and construct homotopy colimits of diagrams of objects in ABC cofibration categories. Homotopy colimits for Quillen model categories are obtained as a particular case. We attach to each ABC cofibration category a left Heller derivator. A dual theory is developed for homotopy limits in ABC fibration categories and for right Heller derivators. These constructions provide a natural framework for 'doing homotopy theory' in ABC (co)fibration categories.
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math.AT 1years
2026 1verdicts
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Homotopy theories via the magnitude-path spectral sequence
Defines r-quasi-isomorphisms and r-cofibrations on generalized metric spaces so that each page of the magnitude-path spectral sequence satisfies metric Eilenberg-Steenrod axioms and supports Brown category structures for homotopy colimits, restricting to directed graphs at r=1.