Related papers: Six-Functor Formalisms II : The $\infty$-categoric…

We show that the functor sending a locally compact Hausdorff space $X$ to the $\infty$-category of spectral sheaves $\mathrm{Shv}(X; \mathrm{Sp})$ is initial among all continuous six-functor formalisms on the category of locally compact…

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…

Context: Reynolds showed us how to use continuation-passing style and defunctionalization to transform a recursive interpreter for a language into an abstract machine for programs in that language. The same techniques explain other…

After the universal property of the six functor formalism $\Shv(-;\Sp)$ on locally compact Hausdorff spaces given by Zhu, we show that the six functor formalism $\Shv(-;\Sp)$ on light condensed anima in the sense of Heyer-Mann is initial…

We characterize $t$-structures in stable $\infty$-categories as suitable quasicategorical factorization systems. More precisely we show that a $t$-structure $\mathfrak{t}$ on a stable $\infty$-category $\mathbf{C}$ is equivalent to a normal…

We prove a result concerning formality of the pull-back of a fibration. Our approach is to use bar complexes in the category of commutative differential graded algebras. As an application, we generalize an old result of Baum and Smith.

This is a major update of the previous version. The methods of the paper are now fully constructive and the style is "formalization ready" with the emphasis on the possibility of formalization both in type theory and in constructive set…

We describe the supports of a class of real-valued maps on $C*(X)$ introduced by Radul. Using this description, a characterization of compact-valued retracts of a given space in terms of functional extenders is obtained. For example, if…

This article gives an elementary and formal 2-categorical construction of a bicategory of right fractions analogous to anafunctors, starting from a 2-category equipped with a family of covering maps that are fully faithful and co-fully…

We study the framework of $\infty$-equipments which is designed to produce well-behaved theories for different generalizations of $\infty$-categories in a synthetic and uniform fashion. We consider notions of (lax) functors between these…

We study categories of matrix factorizations. These categories are defined for any regular function on a suitable regular scheme. Our paper has two parts. In the first part we develop the foundations; for example we discuss derived direct…

In the seminal work of Gaitsgory and Rozenblyum on derived algebraic geometry, eight conjectures regarding the theory of $(\infty,2)$-categories are stated. This paper aims to clarify the status of these claims, and to provide a proof for…

We introduce two novel complementary notions of the Lefschetz number for a functor from a finite acyclic category to itself and we prove a Lefschetz fixed-object theorem and a Lefschetz fixed-morphism theorem. In order to do so, we use the…

In this paper, we generally describe a method of taking an abstract six functors formalism in the sense of Khan or Cisinski-D\'{e}glise, and outputting a derived motivic measure in the sense of Campbell-Wolfson-Zakharevich. In particular,…

With the aim of developing the concepts of positive logic and in response to a question that was asked by Poizat in one of his articles, I wrote this article. The main topic is the study of compactness in the extension as a compact…

We study problems concerning the Samuel compactification of the automorphism group of a countable first-order structure. A key motivating question is a problem of Furstenberg and a counter-conjecture by Pestov regarding the difference…

We study compactifications of the $N=2$ 6D tensionless string on various complex two-folds down to two-dimensions. In the IR limit they become non-trivial conformal field theories in 2D. Using results of Vafa and Witten on the partition…

Let K be an algebraically closed field, X a K-scheme, and X(K) the set of closed points in X. A constructible set C in X(K) is a finite union of subsets Y(K) for finite type subschemes Y in X. A constructible function f : X(K) --> Q has…

The simplicial extension of any functor from Sets to Sets which commutes with directed colimits takes weak equivalences to weak equivalences. The goal of the present paper is construct a framework which can be used to proof results of this…

We define natural A_infinity-transformations and construct A_infinity-category of A_infinity-functors. The notion of non-strict units in an A_infinity-category is introduced. The 2-category of (unital) A_infinity-categories, (unital)…