Related papers: Hypersheaves and bases

Given a stratified topological space, we answer the question whether the functor from the derived category of constructible sheaves to the derived category of sheaves with constructible cohomology is an equivalence. We also establish basic…

In this note we study symmetric monoidal functors from a symmetric monoidal 1-category to a cartesian symmetric monoidal $\infty$-category, which are in addition hypersheaves for a certain topology. We prove a symmetric monoidal version of…

Here we prove that the homological dimension of the category of sheaves on a topological space satisfying some suitable conditions is finite. In particular, we find conditions to bound the homological dimension of o-minimal and subanalytic…

The purpose of this paper is to explain why the functor that sends a stratified topological space $S$ to the $\infty$-category of constructible (hyper)sheaves on $S$ with coefficients in a large class of presentable $\infty$categories is…

The goal of this paper is to explain when basechange theorems for sheaves of spaces imply basechange for sheaves with coefficients in other presentable $\infty$-categories. We accomplish this by analyzing when the tensor product of…

In view of the fact that many of the most familiar examples of category bases are equivalent to some topology, it is natural to ask whether category bases are always topological in nature. The answer is in the negative. In this paper, we…

We characterise the class of exponentiable $\infty$-toposes: $\mathcal X$ is exponentiable if and only if $\mathcal S\mathrm{h}(\mathcal X)$ is a continuous $\infty$-category. The heart of the proof is the description of the…

Let A and B be normal matrices with coefficients that are continuous complex-valued functions on a topological space X that has the homotopy type of a CW complex, and suppose these matrices have the same distinct eigenvalues at each point…

We define the notion of {\em classifying space} of a topological stack and show that every topological stack \X has a classifying space X which is a topological space well-defined up to weak homotopy equivalence. Under a certain…

We prove, over any base ring, that the infinity-category of strictly unital A-infinity-categories (and strictly unital functors) is equivalent to the infinity-category of unital A-infinity-categories (and unital functors). We also identify…

For a (possibly large) realized limit sketch $\mathcal{S}$ such that every $\mathcal{S}$-model is small in a suitable sense we show that the category of cocontinuous functors $\mathsf{Mod}(\mathcal{S}) \to \mathcal{C}$ into a cocomplete…

Given a suitable functor T:C -> D between model categories, we define a long exact sequence relating the homotopy groups of any X in C with those of TX, and use this to describe an obstruction theory for lifting an object G in D to C.…

We introduce the notion of a geometric $(\infty,1)$-category, the protopyical example of which is an $(\infty,1)$-topos. We study (hyper)sheaves on geometric $(\infty,1)$-categories, proving that these are characterized by a form of…

Cosheaves are a dual notion of sheaves. In this paper, we prove existence of a dual of sheafifications, called \textit{cosheafifications}, in the $\infty$-category theory. We also prove that the $\infty$-category of $\infty$-cosheaves is…

If a Quillen model category can be specified using a certain logical syntax (intuitively, ``is algebraic/combinatorial enough''), so that it can be defined in any category of sheaves, then the satisfaction of Quillen's axioms over any site…

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…

A subunit in a monoidal category is a subobject of the monoidal unit for which a canonical morphism is invertible. They correspond to open subsets of a base topological space in categories such as those of sheaves or Hilbert modules. We…

Given a complex space $X$, we cosidered the problem of finding a {\it hyperbolic model} of $X$. This is an object $\ip(X)$ with a morphism $i:X\to \ip(X)$ in such a way that $\ip(X)$ is ``hyperbolic'' in a suitable sense and $i$ is as close…

Let $X \subset \mathbb{P}^{n+1}$ be a smooth Fano hypersurface of dimension $n$ and degree $d$. The derived category of coherent sheaves on $X$ contains an interesting subcategory called the Kuznetsov component $\mathcal{A}_X$. We show that…

From every pair of adjoint functors it is possible to produce a (possibly trivial) equivalence of categories by restricting to the subcategories where the unit and counit are isomorphisms. If we do this for the adjunction between effect…