Related papers: A convenient differential category

We generalize the notion of a bornology by omitting the condition that a one-point-subset is bounded and obtain a complete and co-complete generalization of the category of bornological coarse spaces. Then we imitate the construction of…

Consider an o-minimal structure on the real field. Let $M$ be a definable $C^r$ manifold, where $r$ is a nonnegative integer. We first demonstrate an equivalence of the category of definable $C^r$ vector bundles over $M$ with the category…

A group-category is an additively semisimple category with a monoidal product structure in which the simple objects are invertible. For example in the category of representations of a group, 1-dimensional representations are the invertible…

Let $X$ be a projective and smooth variety over an algebraically closed field $k$. Let $f:Y\rightarrow X$ be a proper and surjective morphism of $k$-varieties. Assuming that $f$ is separable, we prove that the Tannakian category associated…

Differential categories were introduced to provide a minimal categorical doctrine for differential linear logic. Here we revisit the formalism and, in particular, examine the two different approaches to defining differentiation which were…

A "Chen space" is a set X equipped with a collection of "plots" - maps from convex sets to X - satisfying three simple axioms. While an individual Chen space can be much worse than a smooth manifold, the category of all Chen spaces is much…

A convenient bicategory of topological stacks is constructed which is both complete and Cartesian closed. This bicategory, called the bicategory of compactly generated stacks, is the analogue of classical topological stacks, but for a…

Category theory provides a collective description of many arrangements in mathematics, such as topological spaces, Banach spaces and game theory. Within this collective description, the perspective from any individual member of the…

We introduce the notion of inductive category in a model category and prove that it agrees with the Ganea approach given by Doeraene. This notion also coincides with the topological one when we consider the category of (well-) pointed…

Category theory gives a mathematical characterization of naturality but not of canonicity. The purpose of this paper is to develop the logical theory of canonical maps based on the broader demonstration that the dual notions of elements &…

Nonrigid mathematical structures may no longer form usual Eilenberg - Mac Lane categories, but more general ones, as illustrated by pseudo-topologies. A rather general concept of pseudo-topology was used in constructing differential…

Two theorems witnessing the abundance of geometrically trivial strongly minimal autonomous differential equations of arbitrary order are shown. The first one states that a generic algebraic vector field of degree $d\geq 2$ on the affine…

This work contributes to clarifying several relationships between certain higher categorical structures and the homotopy type of their classifying spaces. Bicategories (in particular monoidal categories) have well understood simple…

This is a report on aspects of the theory and use of monoidal categories. The first section introduces the main concepts through the example of the category of vector spaces. String notation is explained and shown to lead naturally to a…

Let \(C\) be a smooth projective curve over an algebraically closed field of characteristic zero. For the moduli space \(N(r,L)\) of stable vector bundles on \(C\) of rank \(r\) with fixed determinant \(L\), we study the group of exact…

Consider the intrinsic fundamental group \`a la Grothendieck of a linear category using connected gradings. In this article we prove that any full convex subcategory is incompressible, in the sense that the group map between the…

We give a rigorous formulation of the intuitive idea that a differentiable map should be thesame thing as a locally, or infinitesimally, linear map: just as a linear map respects the operations of addition and multiplication by scalars ina…

In this paper we develop the theory of quasispaces (for a Grothendieck topology) and of concrete quasitopoi, over a suitable base category. We introduce the notion of f-regular category and of f-regular functor. The f-regular categories are…

Constructing and manipulating homotopy types from categorical input data has been an important theme in algebraic topology for decades. Every category gives rise to a `classifying space', the geometric realization of the nerve. Up to weak…

We tackle the problem of finding a suitable categorical framework for generalized functions used in mathematical physics for linear and non-linear PDEs. We are looking for a Cartesian closed category which contains both Schwartz…