arXiv subjects

Thorger Geiß

Publications and source records attributed to Thorger Geiß.

3 recordsLinked to original sources

arXiv2026

Cocompactness and Presentability

We give a short proof that $κ$-cocompact objects in a presentable category are subterminal. As our main result, we extend this to the setting of presentable $\infty$-categories. A consequence is that an $\infty$-category $\mathcal{C}$ such that both $\mathcal{C}$ and $\mathcal{C}^\mathsf{op}$ are presentable is a small complete lattice, extending a classical theorem of Gabriel-Ulmer. Along the way, we prove a nilpotence result for phantom maps in general pointed presentable $\infty$-categories. Additionally, we show that a strengthening of our main result is equivalent to the existence of a proper class of measurable cardinals.

math.CT↗

arXiv2026

Factorization Systems on $\infty$-Categories: Un/Straightening and Monadicity

We prove two structural results about factorization systems on $\infty$-categories. Firstly, we classify factorization systems on the total spaces of a class of fibrations of $\infty$-categories in terms of factorization systems on the base and the fibers. This will be interpreted as an un/straightening equivalence for $\infty$-categories equipped with a factorization system, using Juran's double $\infty$-categorical framework. Along the way, we develop the general theory of lifting through faithful functors of $n$-uple $\infty$-categories. Secondly, we prove that the forgetful functor from $\infty$-categories equipped with a factorization system to $\infty$-categories is a monadic right adjoint.

math.CT↗

arXiv2026

On Cofiltered Limits of $\infty$-Categories and Adjunctions

We prove that, under mild assumptions, the limit of a cofiltered diagram of $\infty$-categories is a reflective (resp. coreflective) localization of its oplax (resp. lax) limit. This gives rise to a number of exceptional 'stability' results for categorical properties under such limits, in particular one for adjunctions. As an application, we recover a push-pull formula describing certain filtered colimits in the $\infty$-category $\mathrm{Pr}^L$ of presentable $\infty$-categories.

math.CT↗