Related papers: Filter Quotients and Non-Presentable $(\infty,1)$-…
We introduce the basic elements of the theory of parametrized $\infty$-categories and functors between them. These notions are defined as suitable fibrations of $\infty$-categories and functors between them. We give as many examples as we…
We define the tensor product of filtered $A_\infty$-algebras. establish some of its properties and give a partial description of the space of bounding cochains in the tensor product. Furthermore we show that in the case of classical…
By theorems of Carlson and Renaudin, the theory of $(\infty,1)$-categories embeds in that of prederivators. The purpose of this paper is to give a two-fold answer to the inverse problem: understanding which prederivators model…
Let $n_1,\ldots,n_k $ be integers larger than or equal to 2. We characterize linear maps $\phi: M_{n_1\cdots n_k}\rightarrow M_{n_1\cdots n_k}$ such that $${\mathrm rank}\,(\phi(A_1\otimes \cdots \otimes…
By looking at decidable quotients, a sufficient condition is provided to guarantee that (1) the full subcategory of decidable objects of a topos is an exponential ideal and that (2) the classical notion of connectedness for an object $X$…
We show that the tensor product of $\infty$-categories enriched in a suitable monoidal $\infty$-category preserves colimits in each variable, fixing a mistake in an earlier paper of Gepner and the author. We also prove that essentially…
The importance of accessible categories has been widely recognized; they can be described as those freely generated in some precise sense by a small set of objects and, because of that, satisfy many good properties. More specifically…
We prove general adjoint functor theorems for weakly (co)complete $n$-categories. This class of $n$-categories includes the homotopy $n$-categories of (co)complete $\infty$-categories, so these $n$-categories do not admit all small…
We define a family of a (non-principal) ultrafilters on N which are, in a sense, far from P-points. We first under reasonable conditions, prove its existence. In a continuation we shall prove that such a point may exist while no P-point…
We set the foundations of a theory of Grothendieck $(\infty,2)$-topoi based on the notion of fibrational descent, which axiomatizes both the existence of a classifying object for fibrations internal to an $(\infty,2)$-category as well as…
Let f:A-->B be a covering map. We say A has e filtered ends with respect to f (or B) if for some filtration {K_n} of B by compact subsets, A - f^{-1}(K_n) "eventually" has e components. The main theorem states that if Y is a (suitable) free…
We continue the research of the relation $\hspace{1mm}\widetilde{\mid}\hspace{1mm}$ on the set $\beta {\mathbb{N}}$ of ultrafilters on ${\mathbb{N}}$, defined as an extension of the divisibility relation. It is a quasiorder, so we see it as…
The bulk of this paper is devoted to the comparison of several models for the theory of (infinity,2)-categories: that is, higher categories in which all k-morphisms are invertible for k > 2 (the case of (infinity,n)-categories is also…
The filtered derived category of an abelian category has played a useful role in subjects including geometric representation theory, mixed Hodge modules, and the theory of motives. We develop a natural generalization using current methods…
The Grothendieck construction establishes an equivalence between fibrations, a.k.a. fibred categories, and indexed categories, and is one of the fundamental results of category theory. Cockett and Cruttwell introduced the notion of…
The main objective of this paper is to construct a homotopy colimit functor on a category of functors taking values in the model category of quasi-categories.
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…
We study the singularities of varieties obtained as infinitesimal quotients by $1$-foliations in positive characteristic. (1) We show that quotients by (log) canonical $1$-foliations preserve the (log) singularities of the MMP. (2) We prove…
The purpose of this writing is to show that, if we use the definition of elementary $\infty$-topos that has been proposed by Mike Shulman, then the fact that every geometric $\infty$-topos satisfies the required axioms, more specifically…
The aim of this article is to explain a philosophy for applying higher dimensional Seifert-van Kampen Theorems, and how the use of groupoids and strict higher groupoids resolves some foundational anomalies in algebraic topology at the…