Tootfinder

from my link log —
What category theory teaches us about dataframes.
https://mchav.github.io/what-category-theory-teaches-us-about-dataframes/
saved 2026-03-29

Testing some ideas in case my next linear algebra publication needs a photo
(... or category theory?) 🐈

Higher algebra in $t$-structured tensor triangulated $\infty$-categories
Jiacheng Liang
https://arxiv.org/abs/2603.27786 https://arxiv.org/pdf/2603.27786 https://arxiv.org/html/2603.27786
arXiv:2603.27786v1 Announce Type: new
Abstract: We generalize fundamental notions of higher algebra, traditionally developed within the $\infty$-category of spectra, to the broader setting of $t$-structured tensor triangulated $\infty$-categories ($ttt$-$\infty$-categories). Under a natural structural condition, which we call "projective rigidity", we establish higher categorical analogues of Lazard's theorem and prove the existence and universal property of Cohn localizations. Furthermore, we generalize higher almost ring theory to the $ttt$-$\infty$-categorical setting, showing that $\pi_0$-epimorphic idempotent algebras are in natural bijection with idempotent ideals. By exploiting deformation theory, we establish a general \'etale rigidity theorem, proving that the $\infty$-category of \'etale algebras over a fixed connective base is completely determined by its discrete counterpart. Finally, we characterize the moduli of such projectively rigid $ttt$-$\infty$-categories, demonstrating that the presheaf $\infty$-category on the 1-dimensional framed cobordism $\infty$-category serves as the universal projectively rigid $ttt$-$\infty$-category.
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Introducing pixelation with applications
J. Daisie Rock
https://arxiv.org/abs/2603.25432 https://arxiv.org/pdf/2603.25432 https://arxiv.org/html/2603.25432
arXiv:2603.25432v1 Announce Type: new
Abstract: Motivated by the desire for a new kind of approximation, we define a type of localization called pixelation. We present how pixelation manifests in representation theory and in the study of sites and sheaves. A path category is constructed from a set, a collection of "paths" into the set, and an equivalence relation on the paths. A screen is a partition of the set that respects the paths and equivalence relation. For a commutative ring, we also enrich the path category over its modules (=linearize the category with respect to the ring) and quotient by an ideal generated by paths (possibly 0). The pixelation is the localization of a path category, or the enriched quotient, with respect to a screen. The localization has useful properties and serves as an approximation of the original category. As applications, we use pixelations to provide a new point of view of the Zariski topology of localized ring spectra, provide a parallel story to a ringed space and sheaves of modules, and construct a categorical generalization of higher Auslander algebras of type $A$.
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A pretorsion theory for right groups
Alberto Facchini, Carmelo Antonio Finocchiaro
https://arxiv.org/abs/2603.23982 https://arxiv.org/pdf/2603.23982 https://arxiv.org/html/2603.23982
arXiv:2603.23982v1 Announce Type: new
Abstract: Let $S$ be a right group. Then there exist two congruences $\sim$ and $\equiv$ on $S$ such that $S$ is the product of its quotient semigroups $S/{\sim}$ and $S/{\equiv}$, where $S/{\sim}$ is a group and $S/{\equiv}$ is a right zero semigroup. If $E$ is the set of all idempotents of $S$ and we fix an element $e_0\in E$, then the pointed right group $(S,e_0)$ is the coproduct of its pointed subsemigroups $(Se_0,e_0)$ and $(E,e_0)$ in the category of pointed right groups. In general, there is a pretorsion theory in the category of right groups in which the torsion objects are right zero semigroups and the torsion-free objects are groups.
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[2026-03-31 Tue (UTC), 3 new articles found for math.CT Category Theory]
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[2026-03-30 Mon (UTC), no new articles found for math.CT Category Theory]
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[2026-03-27 Fri (UTC), 2 new articles found for math.CT Category Theory]
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[2026-03-26 Thu (UTC), 2 new articles found for math.CT Category Theory]
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[2026-03-25 Wed (UTC), 1 new article found for math.CT Category Theory]
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