Related papers: Univalence in locally cartesian closed infinity-ca…
We construct a new model category presenting the homotopy theory of presheaves on "inverse EI $(\infty,1)$-categories", which contains universe objects that satisfy Voevodsky's univalence axiom. In addition to diagrams on ordinary inverse…
This is both an expository and research paper where we advocate a systematic study of continuous analogues of finite partially ordered sets, convex polytopes, oriented matroids, arrangements of subspaces, finite simplicial complexes, and…
The Univalence Principle is the statement that equivalent mathematical structures are indistinguishable. We prove a general version of this principle that applies to all set-based, categorical, and higher-categorical structures defined in a…
Category theory in homotopy type theory is intricate as categorical laws can only be stated "up to homotopy", and thus require coherences. The established notion of a univalent category (Ahrens, Kapulkin, Shulman) solves this by considering…
We use quantum invariants to define an analytic family of representations for the mapping class group of a punctured surface. The representations depend on a complex number A with |A| <= 1 and act on an infinite-dimensional Hilbert space.…
We prove the conjecture that any Grothendieck $(\infty,1)$-topos can be presented by a Quillen model category that interprets homotopy type theory with strict univalent universes. Thus, homotopy type theory can be used as a formal language…
We show that the number of deformation types of canonically polarized manifolds over an arbitrary variety with proper singular locus is finite, and that this number is uniformly bounded in any finite type family of base varieties. As a…
We present Voevodsky's construction of a model of univalent type theory in the category of simplicial sets. To this end, we first give a general technique for constructing categorical models of dependent type theory, using universes to…
Homotopy type theory is a new branch of mathematics, based on a recently discovered connection between homotopy theory and type theory, which brings new ideas into the very foundation of mathematics. On the one hand, Voevodsky's subtle and…
In this paper we outline a program for the classification of Floer-type theories, (or defining invariants of finite type for families). We consider Khovanov complexes as a local system on the space of knots introduced by V. Vassiliev and…
We construct a univalent universe in the sense of Voevodsky in some suitable model categories for homotopy types (obtained from Grothendieck's theory of test categories). In practice, this means for instance that, appart from the homotopy…
Partially ordered sets (posets) play a universal role as an abstract structure in many areas of mathematics. For finite posets, an explicit enumeration of distinct partial orders on a set of unlabelled elements is known only up to a…
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 show that Martin Hyland's effective topos can be exhibited as the homotopy category of a path category $\mathbb{EFF}$. Path categories are categories of fibrant objects in the sense of Brown satisfying two additional properties and as…
Let $M$ be a compact, connected manifold of positive dimension and let $\mathcal G\leq\textrm{Homeo}(M)$ be \emph{locally approximating} in the sense that for all open $U\subseteq M$ compactly contained in a single Euclidean chart of $M$,…
We study the problem of when triangulated categories admit unique infinity-categorical enhancements. Our results use Lurie's theory of prestable infinity-categories to give conceptual proofs of, and in many cases strengthen, previous work…
We have studied homeomorphisms that satisfy the Poletsky-type inverse inequality in the domain of the Euclidean space. It is proved that the uniform limit of the family of such homeomorphisms is either a homeomorphism into the Euclidean…
We show that closures of families of unitary local systems on quasiprojective varieties for which the dimension of a graded component of Hodge filtration has a constant value can be identified with a finite union of polytopes. We also…
We define an elementary $\infty$-topos that simultaneously generalizes an elementary topos and Grothendieck $\infty$-topos. We then prove it satisfies the expected topos theoretic properties, such as descent, local Cartesian closure,…
We (a) prove that continuous morphisms from locally compact groups to locally exponential (possibly infinite-dimensional) Lie groups factor through Lie quotients, recovering a result of Shtern's on factoring norm-continuous representations…