Related papers: Pyknotic objects, I. Basic notions

We prove that the generalized symplectic capacities recognize objects in symplectic categories whose objects are of the form $(M, \omega)$, such that $M$ is a compact and 1-connected manifold, $\omega$ is an exact symplectic form on $M$,…

We construct explicit minimal models for the (hyper)operads governing modular, cyclic and ordinary operads, and wheeled properads, respectively. Algebras for these models are homotopy versions of the corresponding structures.

This paper continues a geometric study of Harvey's Complex of Curves, whose ultimate goal is to apply the theory of hyperbolic spaces and groups to algorithmic questions for the Mapping Class Group and geometric properties of Kleinian…

We describe new autoequivalences of derived categories of coherent sheaves arising from what we call $\mathbb P^n$-objects of the category. Standard examples arise from holomorphic symplectic manifolds. Under mirror symmetry these…

The work lays the foundations of the theory of changeable sets. In author opinion, this theory, in the process of it's development and improvement, can become one of the tools of solving the sixth Hilbert problem least for physics of…

This is an introduction to the notion of local subgroupoid introduced by the author and R. Brown. It can also serve as an introduction to an application of sheaf theory, and so could be useful to beginners in that theory. The main results…

We revisit sheaves on locales by placing them in the context of the theory of quantale modules. The local homeomorphisms $p:X\to B$ are identified with the Hilbert $B$-modules that are equipped with a natural notion of basis. The…

We outline the main features of the definitions and applications of crossed complexes and cubical $\omega$-groupoids with connections. These give forms of higher homotopy groupoids, and new views of basic algebraic topology and the…

The small object argument is a method for transfinitely constructing weak factorization systems originally motivated by homotopy theory. We establish a variant of the small object argument that is enriched over a cofibrantly generated weak…

This paper is the first in a series of papers in which we define and study a category of "sheaves of $\mathcal Z$-modules on the set of alcoves" that carries important information on the category of representations of semisimple Lie…

We construct a localic groupoid $\mathbb{G}_{KH}$ such that for any locale $X$ the category of compact Hausdorff locales in the topos of sheaves over $X$ is equivalent to a category whose objects are principal $\mathbb{G}_{KH}$-bundles over…

We give a general framework of equivariant model category theory. Our groups G, called Hopf groups, are suitably defined group objects in any well-behaved symmetric monoidal category V. For any V, a discrete group G gives a Hopf group,…

We show that the classifying space of a $p$-local compact group is approximated by a telescope of classifying spaces of $p$-local finite groups. This result has numerous implications, like a Stable Elements Theorem for $p$-local compact…

We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, ``functors between two homotopy theories form a homotopy theory'', or more precisely that the category of such models…

We show that basic homotopical notions such as homotopy sets and groups, connected and truncated maps, cellular constructions and skeleta, etc., extend to the setting of $(\infty,\infty)$-categories, as well as to presentable categories…

Centers of categories capture the natural operations on their objects. Homotopy coherent centers are introduced here as an extension of this notion to categories with an associated homotopy theory. These centers can also be interpreted as…

Localic and realizability toposes are two central classes of toposes in categorical logic, both arising through the Hyland-Johnstone-Pitts tripos-to-topos construction. We investigate their shared geometric features by providing an…

Skein modules are the main objects of an algebraic topology based on knots (or position). In the same spirit as Leibniz we would call our approach "algebra situs." When looking at the panorama of skein modules we see, past the rolling hills…

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…

Many examples of zeta functions in number theory and combinatorics are special cases of a construction in homotopy theory known as a decomposition space. This article aims to introduce number theorists to the relevant concepts in homotopy…