arXiv subjects

Yuki Kato

Publications and source records attributed to Yuki Kato.

9 recordsLinked to original sources

arXiv2025

Tilting equivalence of finite almost derived algebraic cobordism for perfectoid algebras

In this paper, we prove tilting equivalence for the finite almost derived algebraic cobordism spectrum $\mathrm{dMGL}^{a,\rm fin}$ of perfectoid algebras. More precisely, if $V$ is an integral perfectoid valuation ring and $A$ is an integral perfectoid $V$-algebra, then the tilting functor induces a weak equivalence \[ \mathrm{dMGL}^{a,\rm fin}(A) \simeq \mathrm{dMGL}^{a,\rm fin}(A^\flat). \] This invariant is a finite syntomic, derived, and non-$\mathbb{A}^1$-local version of algebraic cobordism, designed to retain infinitesimal deformation data over mixed characteristic bases. To prove the result, we first establish the corresponding finite non-unital statement and isolate a form of excisive approximation for pointed $\infty$-categories, including non-presentable ones. In the locally finitely presentable case, this agrees with the framework of Heuts. We also define approximation functors along natural transformations and apply them to the comparison between periodic algebraic cobordism and homotopy $K$-theory, obtaining Bott periodicity and Gabber rigidity.

math.CT↗

arXiv2024

Non-unital algebra objects of stable symmetric monoidal model categories by Smith ideal theory

This note remarks that the correspondence between non-unital algebras and augmented unital algebras can be derived from Hovey's Smith ideal theory. Applying Smith ideal theory of stable symmetric monoidal model category, we formulate non-unital algebra objects of stable symmetric monoidal model categories and generalize the correspondence between non-unital algebra objects and augmented algebra objects.

math.CT↗

arXiv2023

Nilpotent approximation and completion of $\mathbb{E}_\infty$-algebra objects of stable symmetric monoidal model categories

We develop a nilpotent approximation theory for Smith ideals, extending adic completion for commutative rings to monoid objects in locally presentable symmetric monoidal abelian categories and to $\mathbb{E}_\infty$-algebra objects in stable symmetric monoidal model categories. The main result is a formal completeness theorem: finite generation of a Smith ideal forces completeness of its nilpotent approximation. This gives a categorical analogue of the finite generation completeness phenomenon in classical adic completion, while remaining distinct from ordinary adic completion of quotient rings. As applications, we construct an almost mathematics version of nilpotent approximation and prove a homotopical completeness theorem for weakly compact Smith ideals. We then apply the general theory to motivic spectra. For the canonical morphism from algebraic cobordism to algebraic K-theory, we construct the corresponding K-theoretic nilpotent approximation of algebraic cobordism, prove its homotopical completeness and Bott periodicity, and establish a mod-$\ell$ Gabber rigidity theorem for the analogous approximation of $\mathbf{MGL}/\ell$ by $\mathbb{K}/l$.

math.CT↗

arXiv2023

Algebraic $K$-theory and algebraic cobordism of almost mathematics

Faltings; Gabber and Ramero introduced almost mathematics. In another way, almost mathematics can be characterized bilocalization abelian category of modules mentioned in Quillen's unpublished note. Applying the concept of Quillen's bilocalization to Gabber and Ramero's work, this paper establishes the almost version of algebraic $K$-theory and cobordism. As a result of almost $K$-theory, we prove that in the case an almost algebra containing a field, the almost $K$-theory of the almost algebra is a direct factor of the $K$-theory of the field, implying that almost $K$-theory holds the Gersten property. We clarify that an almost $K$-theory is a $K$-theory spectrum of non-unital firm algebras in the sense of Quillen. Furthermore, we obtain that almost algebraic cobordism holds tilting equivalence on the category of zero-section stable integral perfectoid algebras with finite syntomic topology.

math.KT↗

arXiv2023

Non-unital algebraic $K$-theory and almost mathematics

The Gersten conjecture is still an open problem of algebraic $K$-theory for mixed characteristic discrete valuation rings. In this paper, we establish non-unital algebraic $K$-theory which is modified to become an exact functor from the category of non-unital algebras to the stable $\infty$-category of spectra. We prove that for any almost unital algebra, the non-unital $K$-theory homotopically decomposes into the non-unital $K$-theory the corresponding ideal and the residue algebra, implying the Gersten property of non-unital $K$-theory of the the corresponding ideal.

math.KT↗

arXiv2023

Almost mathematics of pointed symmetric monoidal model categories by Smith ideal theory

This article is a generalization of a result in Quillen's note ``Module theory over non-unital rings'' giving a one-to-one correspondence between bilocalization of abelian categories of modules and idempotent ideals of the base ring. Faltings; Gabber and Ramero established almost mathematics, the same as Quillen's bilocalization of a category of modules by nil modules. In this paper, by using the theory of Smith ideals mentioned in Hovey and Smith, we consider almost mathematics of symmetric monoidal pointed model categories. We prove a weak analogue of the one-to-one correspondence in Quillen.

math.CT↗

arXiv2022

Algebraic cobordism via spans

We define the algebraic cobordism of $\infty$-categories equipped with universal line bundle data as an initial oriented functor in the associated span category. In the standard motivic framework, this recovers the Thom spectrum model established by Voevodsky, Gepner, and Snaith. Furthermore, assuming that the $\infty$-category contains Grassmann objects of all ranks, we prove that the projective bundle formula and the corresponding Chern-class and Whitney-sum identities hold for any oriented functor satisfying the splitting principle property. We apply the span formalism to perfectoid geometry. For perfectoid algebras $R$ with tilt $R^\flat$, we construct perfectoid cobordism, prove tilting equivalences, and compare the arc-local and $v$-local $p$-adic theories.

math.AT↗

arXiv2019

Phase Transitions in Binary Categorization: Evidence for Dual-System Decision Making

We report experiment results on binary categorization of (i) gray color, (ii) speech sounds, and (iii) number discrimination. Data analysis is based on constructing psychometric functions and focusing on asymptotics. We discuss the transitions between two types of subjects' response to stimuli presented for two-category classification, e.g., visualized shade of gray into "light-gray" or "dark-gray." Response types are (i) the conscious choice of non-dominant category, described by the deep tails of psychometric function, and (ii) subjects' physical errors in recording decisions in cases where the category choice is obvious. Explanation of results is based on the concept of dual-system decision making. When the choice is obvious, System 1 (fast and automatic) determines subjects' actions, with higher probability of physical errors than when subjects' decision-making is based on slow, deliberate analysis (System 2). Results provide possible evidence for hotly debated dual-system theories of cognitive phenomena.

physics.med-ph↗

arXiv2017

Motivic classifying $\infty$-topoi and spectral stacks

In this paper, we develop motivic derived algebraic geometry, an enhancement of derived algebraic geometry adapted to the $\mathbb{A}^1$-homotopy theory of Morel and Voevodsky. We construct motivic model categories by imposing descent for a Grothendieck topology and invariance with respect to an interval object, and use them to formulate motivic versions of $\infty$-categories, $\infty$-topoi, and classifying $\infty$-topoi. We then define motivic spectral schemes and motivic spectral Deligne--Mumford stacks in terms of structured motivic $\infty$-topoi. The main result establishes the existence of a motivic stackification functor: a geometric morphism between compatible motivic classifying $\infty$-topoi induces a pullback functor on structured motivic topoi, and this functor admits a left adjoint relative to the underlying motivic $\infty$-topos.

math.CT↗