Related papers: Effectivity of Generalized Double $\infty$-Categor…

The purpose of this article is threefold: Firstly, we propose some enhancements to the existing definition of 6-functor formalisms. Secondly, we systematically study the category of kernels, which is a certain 2-category attached to every…

Many definitions of weak and strict $\infty$-categories have been proposed. In this paper we present a definition for $\infty$-categories with strict associators, but which is otherwise fully weak. Our approach is based on the existing type…

We introduce a notion of generalized Serre duality on a Hom-finite Krull-Schmidt triangulated category $\mathcal{T}$. This duality induces the generalized Serre functor on $\mathcal{T}$, which is a linear triangle equivalence between two…

This article tackles categorical coherence within a two-dimensional generalization of Lawvere's functorial semantics. 2-theories, a syntactical way of describing categories with structure, are presented. From the perspective here afforded,…

The subject of this paper is the higher structure of the strictification adjunction, which relates the two fundamental bases of three-dimensional category theory: the $\mathbf{Gray}$-category of $2$-categories and the tricategory of…

We develop category theory within Univalent Foundations, which is a foundational system for mathematics based on a homotopical interpretation of dependent type theory. In this system, we propose a definition of "category" for which equality…

We generalize the adjunction between the functors $Rf_*$ and $f^!$ of derived categories of quasi-coherent sheaves for proper morphisms $f\colon X \to Y$ of Noetherian schemes to the following situation: Let $f$ be a finite type morphism…

This paper examines $\mathbb{Z}$-graded manifolds as semiformal homogeneity structures, comparing two polynomial filtrations from their local models. In finite dimensions, these are componentwise equivalent, yielding isomorphic graded…

We show that the notion of $(\infty,n)$-limit defined using the enriched approach and the one defined using the internal approach coincide. We also give explicit constructions of various double $(\infty,n-1)$-categories implementing various…

We show that, over an arbitrary commutative ring, the localizations of the categories of dg categories, of cohomologically unital, of unital and of strictly unital $A_\infty$ categories with respect to the corresponding classes of…

We use the terms "$\infty$-categories" and "$\infty$-functors" to mean the objects and morphisms in an "$\infty$-cosmos." Quasi-categories, Segal categories, complete Segal spaces, naturally marked simplicial sets, iterated complete Segal…

In a triangulated symmetric monoidal closed category, there are natural dualities induced by the internal Hom. Given a monoidal functor f^* between two such catgories and adjoint couples (f^*,f_*) and (f_*,f^!), we prove the necessary…

Categorical symplectic geometry is the study of a rich collection of invariants of symplectic manifolds, including the Fukaya $A_\infty$-category, Floer cohomology, and symplectic cohomology. Beginning with work of Wehrheim and Woodward in…

We exhibit an adjunction between a category of abstract algebras of partial functions and a category of set quotients. The algebras are those atomic algebras representable as a collection of partial functions closed under relative…

In this paper we construct classifying localic categories and groupoids for various bundles equipped with logical structure. When these bundles are local homeomorphisms, we recover the localic groupoids that classify geometric theories,…

In this paper we investigate the construction of bicategories of fractions originally described by D. Pronk: given any bicategory $\mathcal{C}$ together with a suitable class of morphisms $\mathbf{W}$, one can construct a bicategory…

Building on work of Derksen-Fei and Plamondon, we formulate a conjectural correspondence between additive and monoidal categorifications of cluster algebras, which reveals a new connection between the additive reachability conjecture and…

We distinguish between faint, weak, strong and strict localizations of categories at morphism families and show that this framework captures the different types of derived functors that are considered in the literature. More precisely, we…

In this work, we investigate an effective method for showing that functors between categories are left adjoints. The method applies to a large class of categories, namely locally finitely presentable categories, which are ubiquitous in…

We characterize all pairs $(\mathcal{A}$,$\mathcal{B})$ of generalized Riemann differences for which $\mathcal{A}$-differentiability implies $\mathcal{B}$-differentiability. Two generalized Riemann derivatives $\mathcal{A}$ and…