Related papers: The six-functor formalism for rigid analytic motiv…
In this paper, we record some foundational results on adic geometry that seem to be missing in the existing literature. Namely, we develop the Proj construction and a theory of lci closed immersions in the context of locally noetherian…
We propose a new formalism for specifying and reasoning about problems that involve heterogeneous "pieces of information" -- large collections of data, decision procedures of any kind and complexity and connections between them. The essence…
To any rigid analytic space (in the sense of Fujiwara-Kato) we assign an $\mathbb{A}^1$-invariant rigid analytic homotopy category with coefficients in any presentable category. We show some functorial properties of this assignment as a…
We prove metric rigidity for complete manifolds supporting solutions of certain second order differential systems, thus extending classical works on a characterization of space-forms. In the route, we also discover new characterizations of…
Ever since the introduction of motivic homotopy theory, as a well-proposed approximation of Grothendieck's dream, algebraic geometers then have the chance to study schemes via a homotopy theory. However topologists also found that lifting…
In this paper we propose two guiding principles that suggest a number of conjectures (some now proved) about various forms of rigidity for moduli spaces arising in algebraic geometry. Such conjectures have group-theoretic, topological and…
We present an innovative approach to dimensional analysis, based on a general representation theorem for complete quantity functions admitting a covariant scalar representation; this theorem is in turn grounded in a purely algebraic theory…
We develop a linear-algebraic framework for dimensional analysis in systems with constraints, particularly when variables are numerous or related by implicit relations so that direct elimination is impractical. By expressing both…
We consider categories of equivariant mixed Tate motives, where equivariant is understood in the sense of Borel. We give the two usual definitions of equivariant motives, via the simplicial Borel construction and via algebraic…
This paper is part of a series of articles in which we reproduce the statements regarding the abstract six-functor formalism developed by Liu-Zheng. In this paper, we prove a theorem, which is an $\infty$-categorical version for defining…
Users of program analyses expect that results change predictably in response to changes in their programs, but many analyses fail to provide such robustness. This paper introduces a theoretical framework that provides a unified language to…
We give an elementary description of the space of formal periods of a mixed motive. This allows for a simplified reformulation of the period conjectures of Grothendieck and Kontsevich-Zagier. Furthermore, we develop a machinery which in…
We investigate connections between Lipschitz geometry of real algebraic varieties and properties of their arc spaces. For this purpose we develop motivic integration in the real algebraic set-up. We construct a motivic measure on the space…
The goal of this paper is to construct trace maps for the six functor formalism of motivic cohomology after Voevodsky, Ayoub, and Cisinski-D\'{e}glise. We also construct an $\infty$-enhancement of such a trace formalism. In the course of…
Taking an algebraic perspective on the basic structures of Rough Concept Analysis as the starting point, in this paper we introduce some varieties of lattices expanded with normal modal operators which can be regarded as the natural rough…
This article is the last of the series of articles where we reprove the foundational ideas of abstract six-functor formalisms developed by Liu-Zheng. We prove the theorem of partial adjoints, which is a simplicial technique of encoding…
This paper outlines a general formal framework for reasoning systems, intended to support future analysis of inference architectures across domains. We model reasoning systems as structured tuples comprising phenomena, explanation space,…
Several formal systems, such as resolution and minimal model semantics, provide a framework for logic programming. In this paper, we will survey the use of structural proof theory as an alternative foundation. Researchers have been using…
Starting from simple and necessary axioms on a (derivator enhanced) four-functor-formalism, we construct derivator six-functor-formalisms using compactifications. This works, for instance, for the stable homotopy categories of…
We present a comprehensive programme analysing the decomposition of proof systems for non-classical logics into proof systems for other logics, especially classical logic, using an algebra of constraints. That is, one recovers a proof…