Related papers: Pullbacks in tangent categories and tangent displa…
Discretizing variational principles, as opposed to discretizing differential equations, leads to discrete-time analogues of mechanics, and, systematically, to geometric numerical integrators. The phase space of such variational…
We investigate when the tangent bundle of a projective manifold has a non-trivial first order (or positive-dimensional) deformation. This leads to a new conjectural characterization of the complex projective space.
For a quasi-compact quasi-separated scheme X and an arbitrary scheme Y we show that the pullback construction implements an equivalence between the discrete category of morphisms Y --> X and the category of cocontinuous tensor functors…
In this paper we consider convex subsets of locally-convex topological vector spaces. Given a fixed point in such a convex subset, we show that there exists a curve completely contained in the convex subset and leaving the point in a given…
Differential categories provide the categorical foundations for the algebraic approaches to differentiation. They have been successful in formalizing various important concepts related to differentiation, such as, in particular,…
A recent general model of entanglement, [5], that goes much beyond the usual one based on tensor products of vector spaces is further developed here. It is shown that the usual Cartesian product can be seen as two extreme particular…
It is given the diffeomorphism classification on generic singularities of tangent varieties to curves with arbitrary codimension in a projective space. The generic classifications are performed in terms of certain geometric structures and…
We define tilting subcategories in arbitrary exact categories to archieve the following. Firstly: Unify existing definitions of tilting subcategories to arbitrary exact categories. Discuss standard results for tilting subcategories:…
This paper has been withdrawn and replaced by arXiv:1309.5035. In this paper we describe some examples of so called spherical functors between triangulated categories, which generalize the notion of a spherical object. We also give…
Drawing parallels with hyperplane arrangements, we develop the theory of arrangements of submanifolds. Given a smooth, finite dimensional, real manifold $X$ we consider a finite collection $\mathcal{A}$ of locally flat, codimension-1…
This survey aims to provide a guide to the literature on topological 4-manifolds. Foundational theorems on 4-manifolds are stated, especially in the topological category. Precise references are given, with indications of the strategies…
The tangent bundle $T^kM$ of order $k$, of a smooth Banach manifold $M$ consists of all equivalent classes of curves that agree up to their accelerations of order $k$. In the previous work of the author he proved that $T^kM$, $1\leq k\leq…
We introduce a notion of stratification for rigidly-compactly generated tensor-triangulated categories relative to the homological spectrum and develop the fundamental features of this theory. In particular, we demonstrate that it exhibits…
Given a right adjoint functor between triangulated categories and an object in the target category, we show that the unit map of adjunction on that object is a split monomorphism if and only if the object belongs to the additive closure of…
The primary interest of this paper is to discuss the role of twisting cochains in the theory of characteristic classes. We begin with the homological description of monodromy map, associated with a connection on a trivial bundle over a…
In this paper, we study a new operation named pushforward on diffeological vector pseudo-bundles, which is left adjoint to the pullback. We show how to pushforward projective diffeological vector pseudo-bundles to get projective…
Several authors have recently constructed characteristic classes for classes of infinite rank vector bundles appearing in topology and physics. These include the tangent bundle to the space of maps between closed manifolds, the infinite…
We propose a new notion of `n-category with duals', which we call a Whitney n-category. There are two motivations. The first is that Baez and Dolan's Tangle Hypothesis is (almost) tautological when interpreted as a statement about Whitney…
Manifold hypotheses are typically used for tasks such as dimensionality reduction, interpolation, or improving classification performance. In the less common problem of manifold estimation, the task is to characterize the geometric…
We consider the internalization of the usual notion of principal bundle in a site that has all pullbacks and a terminal object. We use this notion to consider the explicit construction of quotient prestacks via presheaves of categories of…