Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1909.07620

Enriched categories and tropical mathematics

Soichiro Fujii
View original source ↗Read source document ↗Export metadata ↓

Abstract

This is a survey paper on the connection of enriched category theory over a quantale and tropical mathematics. Quantales or complete idempotent semirings, as well as matrices with coefficients in them, are fundamental objects in both fields. We first explain standard category-theoretic constructions on matrices, namely composition, right extension, right lifting and the Isbell hull. Along the way, we review known reformulations (due to Elliott and Willerton) of tropical polytopes, directed tight spans and the Legendre--Fenchel transform by means of these constructions, illustrating their ubiquity in tropical mathematics and related fields. We then consider complete semimodules over a quantale $\mathcal{Q}$, a tropical analogue of vector spaces over a field, and mention Stubbe's result identifying them with skeletal and complete $\mathcal{Q}$-categories. With the aim to bridge a gap between enriched category theory and tropical mathematics, we assume no knowledge in either field.

Explore related subjects

math.CTmath.MG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Soichiro Fujii. 2020-05-18. Enriched categories and tropical mathematics. https://arxiv.org/abs/1909.07620

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Hochschild cohomology of the second kind: Koszul duality and Morita invariance

We define Hochschild cohomology of the second kind for differential graded (dg) or curved algebras as a derived functor in the twisted derived category. The Hochschild cohomology of the second kind of a curved or curved algebra $A$ is then equivalent to the classical Hochschild cohomology of the twisted derived dg category of $A$, which is often geometrically meaningful. Examples include the category of $\infty$-local systems on a topological space, the bounded derived category of a complex manifold and the category of matrix factorizations. We also show that Hochschild cohomology of the second kind is preserved under (nonconilpotent) Koszul duality and weak equivalences of curved algebras. The main technical ingredient is a new bimodule version of Koszul duality.

math.CT↗
arXiv2026

Localization of lax symmetric monoidal categories

In this note, we explain in some detail how one can fiberwise localize a (co)lax symmetric monoidal infinity-category. This construction was tacitly used in Section 5 of our recent paper "On the equivalence of the Lurie's infinity-operads and dendroidal infinity-operads". Version 2: A stronger version of the result is proven. Given a locally cocartesian fibration $f:X\to B$ and a collection of marked arrows $X^\circ\subset f^{-1}(B^{eq})$ in $X$ closed under locally cocartesian liftings, we prove that the localization $\mathcal{L}(X,X^\circ)\to B$ is also a locally cocartesian fibration whose fibers are localizations of the fibers of $f$. This result is applied to the description of localizations of lax symmetric monoidal categories.

math.CT↗
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↗