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Integrated Article


Doctor of Philosophy




Christensen, John D.


We develop the theory of reflective subfibrations on an ∞-topos E. A reflective subfibration L on E is a pullback-compatible assignment of a reflective subcategory D_X ⊆ E/X with associated localization functor L_X, for every X in E. Reflective subfibrations abound in homotopy theory, albeit often disguised, e.g., as stable factorization systems. The added properties of a reflective subfibration L on E compared to a mere reflective subcategory of E are crucial for most of our results. For example, we can prove that L-local maps (i.e., those maps p in D_X for some X in E) admit a classifying map. The existence of such a classifying map is a powerful tool that we exploit to show that there is a reflective subfibration L' whose local maps are exactly the L-separated maps, that is, those maps with L-local diagonal. We investigate some interactions between L and L' and explain when the two reflective subfibrations coincide. Finally, we show the existence of reflective subfibrations associated to sets of maps in E and describe some of their properties.

Summary for Lay Audience

In Mathematics, one often faces two important needs.

  1. Simplify some problems, by discriminating between the properties that are relevant and those that are ancillary for a certain issue.
  2. Present some mathematical objects of interest in different ways, so as to highlight different aspects of these objects.
A powerful mathematical tool to answer these needs is localization theory,
which singles out the local properties and objects that are pertinent to the study of
a problem. Localization theory is particularly useful when studying spaces. Classically,
this study was carried out geometrically in homotopy theory, a branch of
Mathematics that classifies spaces by looking at whether or not one can be
deformed into the other. More recently, localizations of spaces have been studied
logically in homotopy type theory, a syntatic language for reasoning formally
about spaces. This works merges the latter approach with the former, providing a new framework
for understanding spaces through their localizations. The most important features
of our approach are: a simultaneous treatment of the localization of both spaces
and maps between them; and the usage of a modern language that allows the
abstraction of the notion of ``space" to mean anything that can be thought of as
having points, paths between these points, paths between these paths, and so on.
In our work, we recover all classically studied examples of localizations of spaces,
and give new insights to their properties. In bridging the logical and the geometrical approaches to localizations of spaces,
we establish a dictionary between the two approaches that can help in evaluating
the advantages and disadvantages of both, and merging the two communities

Creative Commons License

Creative Commons Attribution 4.0 License
This work is licensed under a Creative Commons Attribution 4.0 License.