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Anders Kock

Publications and source records attributed to Anders Kock.

At least 19 recordsLinked to original sources
arXiv2021

Square-densities, and volume forms

Heron's formula from antiquity, for the area of a triangle, is used to relate volume form and infinitesimal square-volume of certain infinitesimal simplices in a Riemannian manifold

math.DG↗
arXiv2019

Integration of 1-forms and connections

We give a combinatorial/geometric argument of the classical result that an affine connection, which is both torsion free and curvature free, is locally an affine space.

math.DG↗
arXiv2018

Column symmetric polynomials

We study the polynomial algebra (over a ring containing the rationals) in an n by m matrix of variables, and subject to the relation that says that the product of any two variables in the same column is zero. We show that the sub-algebra of polynomials, which are invariant under n! permutations of the columns, is a quotient of the polynomial algebra in m variables; the quotient map sends the ith variable to the sum of the entries in the ith row of the matrix. - An application in synthetic differential geometry is sketched

math.AC↗
arXiv2018

Huygens' principle - a synthetic account

We present an axiomatic/synthetic account of the Huygens Principle of wave fronts. The primitive notions are "touching", and (a weak notion of ) metric. The paper simplifies some of the exposition of the author's "Metric spaces and SDG", Theory and Appl. of Categories 32 (2017), 803-822

math.DG↗
arXiv2017

Metric spaces and SDG

We explore how the synthetic theory of metric spaces (Busemann) can coexist with synthetic differential geometry in the sense based on nilpotent elements in the number line.

math.MG↗
arXiv2015

Affine combinations in affine schemes

We prove that finite sets of mutual neighbor points in an affine scheme admit affine combinations, preserved by any map. Furthermore, such combination has a value which is neighbor point of all the original points.

math.AG↗
arXiv2015

Bundle functors and fibrations

We give an account, in terms of fibered categories and their fibrewise duals, of aspects of the theory of bundle functors and star-bundle functors in differential geometry.

math.CT↗
arXiv2015

The dual fibration in elementary terms

We give an elementary construction of the dual fibration of a fibration. It does not use the non-elementary notion of (pseudo-) functor into the category of categories.

math.CT↗
arXiv2014

Duality for generic algebras

We prove that double dualization into the generic algebra for an algebraic theory has some Gelfand- or Stone- duality properties

math.CT↗
arXiv2013

Fibrations as Eilenberg-Moore algebras

We give an elementary exposition of some fundamental facts about fibered (or rather opfibered) categories, in terms of monads and 2-categories. The account avoids any mention of category-valued functors and pseudofunctors.

math.CT↗
arXiv2013

Projective lines as groupoids with projection structure

The coordinate projective line over a field is seen as a groupoid with a further `projection' structure. We investigate conversely to what extent such an, abstractly given, groupoid may be coordinatized by a suitable field constructed out of the geometry.

math.CT↗
arXiv2012

Local fibered right adjoints are polynomial

For any locally cartesian closed category E, we prove that a local fibered right adjoint between slices of E is given by a polynomial. The slices in question are taken in a well known fibered sense.

math.CT↗
arXiv2011

Commutative monads as a theory of distributions

The theory of commutative monads on cartesian closed categories provides a framework where aspects of the theory of distributions and other extensive quantities can be formulated and some results proved. We make explicit a link between our theory and the theory of Schwartz distributions of compact support. We also discuss probability distributions.

math.CT↗
arXiv2011

Calculus of extensive quantities

We show how a commutative monad gives rise to a theory of extensive quantities, including (under suitable further conditions) a differential calculus of such. The relationship to Schwartz distributions is dicussed. The paper is a companion to the author's "Monads and extensive quantities", but is phrased in more elementary terms.

math.CT↗
arXiv2011

Monads and extensive quantities

If T is a commutative monad on a cartesian closed category, then there exists a natural T-bilinear pairing from T(X) times the space of T(1)-valued functions on X ("integration"), as well as a natural T-bilinear action on T(X) by the space of these functions. These data together make the endofunctors T and "functions into T(1)" into a system of extensive/intensive quantities, in the sense of Lawvere. A natural monad map from T to a certain monad of distributions (in the sense of functional analysis (Schwartz)) arises from this integration.

math.CT↗