arXiv subjects

Kevin Arlin

Publications and source records attributed to Kevin Arlin.

4 recordsLinked to original sources

arXiv2024

The diagrammatic presentation of equations in categories

Lifts of categorical diagrams $D\colon\mathsf{J}\to\mathsf{X}$ against discrete opfibrations $\pi\colon\mathsf{E}\to\mathsf{X}$ can be interpreted as presenting solutions to systems of equations. With this interpretation in mind, it is natural to ask if there is a notion of equivalence of diagrams $D\simeq D'$ that precisely captures the idea of the two diagrams "having the same solutions''. We give such a definition, and then show how the localisation of the category of diagrams in $\mathsf{X}$ along such equivalences is isomorphic to the localisation of the slice category $\mathsf{Cat}/\mathsf{X}$ along the class of initial functors. Finally, we extend this result to the 2-categorical setting, proving the analogous statement for any locally presentable 2-category in place of $\mathsf{Cat}$.

math.CT↗

arXiv2019

Detecting isomorphisms in the homotopy category

We show that the homotopy category of unpointed spaces admits no set of objects jointly reflecting isomorphisms by giving an explicit counterexample involving large symmetric groups. We also show that, in contrast, the spheres jointly reflect equivalences in the homotopy 2-category of spaces. The non-existence of such a set in the homotopy category was originally claimed by Heller, but his argument relied on the statement that for every set of spaces, long enough transfinite sequential diagrams admit weak colimits which are privileged with respect to the given set. Using the theory of graphs of groups, we show that this statement is false, by proving that for every ordinal with uncountable cofinality, there is a diagram indexed by that ordinal which admits no weak colimit that is privileged with respect to the spheres.

math.AT↗

arXiv2018

On Whitehead's theorem beyond pointed connected spaces

We prove that the 2-category of spaces admits a strong generator made up of the tori. In other words, Whitehead's theorem holds for the 2-category of (not necessarily connected, not pointed) spaces.

math.AT↗

arXiv2016

On the $\infty$-categorical Whitehead theorem and the embedding of quasicategories in prederivators

We show that small quasicategories embed, both simplicially and 2-categorically, into prederivators defined on arbitrary small categories, so that in some senses prederivators can serve as a model for $(\infty,1)$-categories. The result for quasicategories that are not necessarily small, or analogously for small quasicategories when mapped to prederivators defined only on finite categories, is not as strong. We prove, instead, a Whitehead theorem that prederivators (defined on any domain) detect equivalences between arbitrarily large quasicategories.

math.CT↗