Research, blogs, and papers
Here you can find my blog posts and an archive of my papers, talking about my research and topics that I have
come across in the course of my research. There is currently not much present, but hopefully that will change in
due time. At least, it will change if I actually dedicate myself to producing content for this website.
Hub
Blog
Methods akin to those of set foundations in arbitrary categories
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dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex
ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat
nulla pariatur.
Originally published June 30, 2024 at 8:00 AM
Papers
The interpretation of powersets in the context of (0,1)-category theory ands its applications to point-set
topology
Given a preordered set \((X, \leq_X)\) we may equip \(X\) with a natural topology called the
Alexandrov topology , where the upper subsets of the preordered set are declared to be the open
subsets of the induced topology. This construction give a covariant functor from from the category of
preordered sets with monotone maps as morphisms to the category of topological spaces.
In an alternative perspective, preordered sets are precisely (0,1)-categories. The points of \(X\) are
its objects, and the \(\leq_X\) relationship between points are the morphisms in \(X\). From this
point-of-view, the Alexandrov topology takes a category and assigns it to a topological space where the
objects are our points and the open sets respect the arrows between them. We discuss this perspective, and
ways to understand common, canonical constructions in the category of topological spaces by viewing them
through the lens of the Alexandrov topology of a suitable preordered set. We also discuss implications for
higher category theory (most specifically, the theory of 1-categories) and also how the various separation
axioms for topological spaces may be understood from this perspective.
Originally published June 24, 2024 at 9:30 AM