Metrically generated theories

Colebunders, E.; Lowen, R.

doi:10.1090/S0002-9939-04-07633-6

Metrically generated theories
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by E. Colebunders and R. Lowen PDF

Proc. Amer. Math. Soc. 133 (2005), 1547-1556 Request permission

Abstract:

Many examples are known of natural functors $K$ describing the transition from categories $\mathcal {C}$ of generalized metric spaces to the “metrizable" objects in some given topological construct $\mathcal {X}$. If $K$ preserves initial morphisms and if $K(\mathcal {C})$ is initially dense in $\mathcal {X}$, then we say that $\mathcal {X}$ is $\mathcal {C}$-metrically generated. Our main theorem proves that $\mathcal {X}$ is $\mathcal {C}$-metrically generated if and only if $\mathcal {X}$ can be isomorphically described as a concretely coreflective subconstruct of a model category with objects sets structured by collections of generalized metrics in $\mathcal {C}$ and natural morphisms. This theorem allows for a unifying treatment of many well-known and varied theories. Moreover, via suitable comparison functors, the various relationships between these theories are studied.

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