Geometric algebra of projective lines
The projective line over a field carries structure of a groupoid with a certain correspondence between objects and arrows. We discuss to what extent the field can be reconstructed from the groupoid.
math.AG↗
arXiv subjects
Anders Kock
Publications and source records attributed to Anders Kock.
Abstract Projective Lines
We describe a notion of (abstract) projective line over a field as a set equipped with a certain first order structure, and a projectivity between projective lines as a bijection preserving this structure. The structure in question is that of a groupoid, with certain properties. This leads to a natural notion of bundle of projective lines, forming a stack.
math.AG↗
Infinitesimal cubical structure, and higher connections
In the context of Synthetic Differential Geometry, we describe a notion of higher connection with values in a cubical groupoid. We do this by exploiting a certain structure of cubical complex derived from the first neighbourhood of the diagonal of a manifold. This cubical complex consists of infinitesimal parallelelpipeda.
math.CT↗
Some matrices with nilpotent entries, and their determinants
We study algebraic properties of matrices whose rows are mutual neighbours, and are also neigbours of 0 ("neighbour" in the sense of a certain nilpotency condition). The intended application is in synthetic differential geometry. For a square matrix of this kind, the product of the diagonal entries equals the determinant, modulo a factor n!
math.RA↗
Commutation Structures
For a fixed object X in a monoidal category, an X-commutation structure on an object A is just a map from XA to AX. We study aspects of such structure in case A has a dual.
math.CT↗
Pregroupoids and their enveloping groupoids
We prove that the forgetful functor from groupoids to pregroupoids has a left adjoint, with the front adjunction injective. Thus we get an enveloping groupoid for any pregroupoid. We prove that the category of torsors is equivalent to that of pregroupoids. Hence we also get enveloping groupoids for torsors, and for principal fibre bundles.
math.CT↗
Categorical distribution theory; heat equation
The aim of this paper is to develop the theory of distributions, not necessarily of compact support, in a topos model of Synthetic Differential Geometry, the so-called "Cahiers Topos". As an application, we study the evolution through time of a heat distribution. In the first part of the paper we study distributions in the sense of Schwartz in the context of convenient vector spaces. In particular, we study smoothness (with respect to time) of the fundamental distribution solution of the heat equation. In the second part, we show how the results of the first part may be used to internalize these notions to prove the existence of the fundamental distribution solution of the heat equation in the Cahiers topos.}
math.CT↗
A geometric theory of harmonic and semi-conformal maps
We describe for any Riemannian manifold a certain infinitesimal neighbourhood of the diagonal. Semi-conformal maps are analyzed as those that preserve such neighbourhoods; harmonic maps are analyzed as those that preserve mirror image formation for pairs of points in such neighbourhoods.
math.DG↗
Some calculus with extensive quantities: wave equation
We take some first steps in providing a synthetic theory of distributions. In particular, we are interested in the use of distribution theory as foundation, not just as tool, in the study of the wave equation.
math.CT↗
First neighbourhood of the diagonal, and geometric distributions
We describe the geometric notion of distribution in synthetic terms, utilizing the notion of "first neighbourhood of the diagonal" from algebraic geometry. We characterize involutive distributions in combinatorial terms.
math.DG↗
Some differential equations in synthetic differential geometry
Some differential equations are considered in the context of Synthetic Differential Geometry. Here, this means that not only nilpotent infinitesimals, but also the formation of function spaces, is exploited. In particular, we utilize distribution spaces in our study of wave and heat equations.
math.CT↗
Infinitesimal aspects of the Laplace operator
In the context of synthetic differential geometry, we study the Laplace operator an a Riemannian manifold. The main new aspect is a neighbourhood of the diagonal, smaller than the second neighbourhood usually required as support for second order differential operators. The new neighbourhood has the property that a function is affine on it if and only if it is harmonic.
math.CT↗
Volume form as volume of infinitesimal simplices
In the context of Synthetic Differential Geometry, we describe the square volume of a ``second-infinitesimal simplex'', in terms of square-distance between its vertices. The square-volume function thus described is symmetric in the vertices. The square-volume gives rise to a characterization of the volume form in the top dimension.
math.CT↗
Algebra of Principal Fibre Bundles, and Connections
We put together some of the efforts by several people of making aspects of fibre bundle theory into algebra. The initiator of these efforts was Charles Ehresmann, who put the notion of groupoid and groupoid action in the focus of fibre bundle theory in general, and in connection theory in particular.
math.CT↗