Homepage of

Anders Kock

At the Department of Mathematical Sciences of Aarhus University .
Office: Building 1530, 2.31
e-mail kock (at) imf.au.dk or kock (at) math.au.dk

Areas of particular interest:

Category Theory, categorical logic.
Applications of categories in differential geometry (synthetic differential geometry, differentiable groupoids).
Topos theory. Two-dimensional categories.

This homepage is a (partly commented) download list. It is not intended to be complete. This in particular applies to items prior to the age of TeX, say prior to 1989. - The dividing lines form an attempt to sort the items according to the decade of (pre-) publication.

Abbreviations for journal names:
TAC: Theory and Applications of Categories
Cahiers: Cahiers de Topologie et Geometrie Differentielle Categorique
JPAA: Journal of Pure and Applied Algebra.

Theory of characteristics of first order partial differential equations (arXiv 2010). Paraphrasing some of the classical Monge-Lie-Klein theory. This is a revised and abbreviated (November 2022) version of my arXiv preprint 1011.5586 from 2010. (The version longer down (2010) on this download list is the revised version.)

Square-volumes, and volume forms: Heron's formula Heron's formula from antiquity is used to relate volume form and infinitesimal square-volume of certain infinitesimal simplices in a Riemannian manifold. To appear in Cahiers (Corrected version of arXiv 2012.06210v2) - Slides available at slides

Barycentric calculus, and the log-exp bijection Slides from an (online) talk at a conference in Banff June 2021 on tangent categories. The talk explains the first order neighbourhood between morphisms in the category of rigs, and the affine combinations that may be formed between such.

Elementwise semantics in categories with pull-backs We extend the use of (``Kripke-Joyal'')- reasoning in categories admitting pull-backs. The aim is to give a theory of jets in this context. In arXiv 2004.14731

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. In arXiv 1902.11003

Column symmetric polynomials (joint with E. Dubuc) We give a description (modulo a certain ideal) of the polynomials in a matrix of variables, invariant under permutation of the columns. A motivating example from SDG is presented. - Cahiers 60 (2019), 241-254.

Affine combinations in affine schemes We show how to form affine combinations of p-tuples of mutually neighbouring points in an affine scheme. In Cahiers 58 (2017),

Bundle functors and fibrations, Tbilisi Math. J. 10(3) (2017), 65-82. We give an account of bundle-functors and star-bundle-functors (known from differential geometry) in terms of fibered categories.

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. The simple axiomatics used implies a synthetic proof of Huygens' principle of wave fronts, as envelopes of a family of spheres. In TAC 2017. -- A simplified account (with emphasis on the contact-element (wave-front) viewpoint) may be found in

Huygens' principle - a synthetic account . (On arXiv 1804.05649.)

New methods for old spaces: synthetic differential geometry A survey talk on the foundations of Synthetic Differential Geometry, Sept. 2015, given at the Workshop New Spaces in Mathematics and Physics, Institut Henri Poincare, Paris, Sept. 28-Oct. 2, 2015. To appear in a forthcoming Proceedings of the workshop, , ed. M. Anel and G. Catren

Duality for generic algebras, Cahiers 56 (2015), 2-14. This is a completed version of an announcement made in 1980, which e.g. implies that the generic algebra R for an algebraic theory T has the property that in the classifying topos for T-algebras, R^R is internally the free R-algebra in one generator, and also a Gelfand type duality for representables.

Projective lines as groupoids with projection structure TAC Vol. 29, (2014), 371-388. 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.

Local fibered right adjoints are polynomial (joint with J. Kock), Math. Structures in Computer Science 23 (2013), 131-141. 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.

Commutative monads as a theory of distributions TAC Vol. 26 (2012), 97-131. We show how the theory of commutative monads, as I developed in the early 1970s, gives a model for a theory of extensive (dsitributed) quantities.

Commutative monads, distributions, and differential categories Constructing differental categories from suitable commutative monads. This is a manuscript for a talk given in Cambridge in 2012, with minor corrections July 2021.

Affine connections, midpoint formation, and point reflection, Theoretical Computer Science 412 (2011), 4770-4777.

Abstract projective lines Cahiers 51 (2010), 224-240.

Cubical version of combinatorial differential forms Appl.Categor Struct. (2010) 18:165-183. Such version depends on the possibility of forming affine combinations of mutually neighbour points

Synthetic Geometry of Manifolds, Cambridge Tracts in Mathematics 180 (2010). A preliminary version (proofread August 7, 2009) of is available (1.9 MB). Cambridge University Press has exclusive copyright (of the final version), so please do not circulate this preliminary version.

Theory of characteristics of first order partial differential equations (arXiv 2010). Paraphrasing some of the classical Monge-Lie-Klein theory. This is a revised and abbreviated version (November 2022) my arXiv preprint 1011.5586 from 2010

Infinitesimal cubical structure and higher connections (arXiv 2007). In the context of Synthetic Differential Geometry, we describe a notion of higher connection with values in a cubical groupoid.

Principal bundles, groupoids, and connections Banach Center Publications 76 (2007), 185-200. This is a summary of some of my work on the issues mentioned in the title, and their relationship through the notion of pregroupoid.

Envelopes - notion and definiteness, Beitraege zur Alg. und Geometrie 48 (2007), 345-350. We examine critically and in terms of Synthetic Differential Geometry, the theory of envelope of a 1-parameter family of surfaces in 3-space.

Ordinary differential equations and their exponentials (joint with G.E. Reyes) Central European J. of Math. 4 (2006), 64-81. We indicate how vector fields on a pair M, N of objects (manifolds, say) give rise to a vector field on the function space [M,N] (the exponential object). Applications are given.

Synthetic Differential Geometry Second Edition , London Math. Soc. Lecture Notes Series 333 (2006), Cambridge University Press. The First Edition appeared as London Math. Soc. Lecture Notes Series 51 (1981), Cambridge University Press. The two editions are identical except in typography, and added historical notes in the Second Edition.

Pregroupoids and their enveloping groupoids Aarhus Math. Preprint Series 2005 no. 3. Also on arXiv

Connections and path connections in groupoids Aarhus Math. Preprint Series 2006 no. 10.

Categorical distribution theory: heat equation (with G.E. Reyes) in arXiv:math/0104164[mathCT].

A geometric theory of harmonic and semi-conformal maps Central European Journal of Mathematics 2(5) 2004 708-724 . Presented at the 5 Krynica Conference on Geometry and Topology of Manifolds, April-May 2003.

Some calculus with extensive quantities: wave equation (with G.E.Reyes), TAC 11 (2003) No. 14. Distributions are here seen as the foundation for studying e.g. the wave equation; distributions are extensive quantities and behave covariantly, unlike functions (densities). In our approach, none of our distributions are assumed to have density functions.

The stack quotient of a groupoid, Cahiers 44 (2003), 85-104.

First neighbourhood of the diagonal, and geometric distributions, Universitatis Iagellonicae Acta Mathematica 41 (2003). This contains a synthetic version of the Ambrose-Singer theorem about holonomy of connections in principal fibre bundles.

Differential calculus and nilpotent real numbers, Bulletin of Symbolic Logic 9 (2003), 225-230.

Differential Forms as Infinitesimal Cochains. Journ. Pure Appl. Algebra 154 (2000), 257-264. A simplicial map from the de Rham complex to the singular complex of a manifold is provided. In particular, wedge product of differential forms is already on the cochain level seen as identical to cup product of singular cochains.

Some differential equations in SDG (with G.E. Reyes) in arXiv:math/0104164[mathCT]. Most of this is subsumed in our two papers on Wave Equation, and on Heat Equation.

Infinitesimal aspects of the Laplace operator, TAC 9 (2001). 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 seen as support for second order differential operators.

Characteriztion of stacks of principal fibre bundles, Institut Mittag-Leffler, Report 2000/2001 No. 27 (2001). - Principal fibre bundles here means: torsors over a groupoid.

The osculating plane of a space curve - synthetic formulations, Rend.Circ.Mat. Palermo II Vol. 64 (2000), 67-79. This proves a well known result in of differential geometry by purely synthetic means, meaning that no coordinatization of any kind appears.

Aspects of Fractional Exponent Functors (with G.E.Reyes), TAC, Vol. 5 (1999), No. 10. Fractional exponents come from amazing right adjoints/atoms in the sense of Lawvere, and are here used in conjunction with enriched category theory to provide a proof of a Theorem of Lawvere on toposes of differential equations.

A note on frame distributions, (with G.E. Reyes), Cahiers 40 (1999), 127-140. A frame distribution is a sup preserving map from a frame in a topos to its subobject classifier. We comment on such as an extensive quantity, partially following Bunge, Funk, and Lawvere.

Geometric Construction of the Levi-Civita Parallelism, Theory and Applications of Categories, Vol. 4 (1998), No.9. This describes the notion of Riemannian metric in terms of a square distance function on the second neighbourhood of the diagonal. The parallelism is constructed by a variational principle.

Combinatorics of curvature, and the Bianchi identity, TAC, Vol 2 (1996), 69-89.

Glueing analysis for complemented subtoposes, (with T. Plewe) TAC, Vol. 2 (1996), 100–112.

Spaces with local equivalence relations, and their monodromy (with I. Moerdijk), Topology and its Applications 72 (1996) 47-78.

Monads for which structures are adjoint to units , Journ. of Pure and Appl. Algebra 104 (1995), 41-59. This is one of several of papers I have written with this title, the first is an Aarhus Preprint 1972/73 No. 35. They deal with what is now often called "KZ-monads". T The version from Feb. 1992 is the most algebraic of the versions; it appeared as an Aarhus Preprint, and appears recompiled here.

Relatively Boolean an de Morgan toposes and locales, (with G.E. Reyes), Cahiers 35 (1994), 249-261.

Generators and Relations for Delta as a Monoidal 2-Category , Beiträ ge zur Algebra und Geometrie 34 (1993), 201-208. It shows that Delta contains a generic KZ monad

Every etendue comes from a local equivalence relation (with I. Moerdijk), Journal of Pure and Applied Algebra 82 (1992) 155-174

Presentation of etendues (joint with I. Moerdijk) Cahiers 32 (1991), 145-164. We prove that every etendue may be presented by a site all of whose maps are monics.

Algebras for the Partial Map Classifier Monad, in Carboni, Pedicchio and Rosolini (eds.) Category Theory. Proceedings Como 1990. Springer Lecture Notes in Math. 1488 (1991), 262-278.

Postulated colimits and left exactness of Kan Extensions Aarhus Preprint 1989/90 no. 9, Retyped in TeX in the fall of 2003.

A coherent theory of sites (with J. Schmidt), Bulletin de la Soc. Math. de Belgique (Serie A), 41 (1989), 321-331. We describe in coherent (= finitary geometric) language a notion of site.

Fibre bundles in general categories 1989, JPAA 56 (1989), 233-245.

A Godement Theorem for locales, Math. Proc. Cambridge Phil. Soc. 105 (1989), 463-471. doi.org/10.1017/S0305004100077835

Mathematical structure of physical quantities, Archive for Rational Mechanics and Analysis 107 (1989), 99-104.

A note on closed ideals in rings of smooth functions (with M. Adelman), Monatshefte fur Mat. 107 (1989), 1-3.

On the Integration Theorem for Lie Groupoids, Czechoslovak Math. J. 39 (114), 1989.

Convenient vector spaces embed into the Cahiers topos Cahiers de topologie et geometrie differentielle categoriques 27 (1986), 3-17.

Corrigendum and addenda to Convenient vector spaces embed .. (with G.E. Reyes), Cahiers de topologie et geometrie differentielle categoriques 28 (1987), 99-110.

Generalized fibre bundles, in Categorical Algebra and its Applications, Louvain la Neuve 1987 (ed. F. Borceux) Springer Lecture Notes in Math. 1348, 194-207.

Lie group valued integration in well adapted toposes , Bull. Austral. Math. Soc. 34 (1986), 395-410

Synthetic reasoning in differential geometry, Revista Colombiana de Mat. 20 (1986), 129-146.

Combinatorial notions relating to principal fibre bundles, JPAA 39 (1986), 141-151.

Infinitesimal deformations of complete vector field are complete retyping of Aarhus Math. Preprint 1985/86 No. 23 (February 1986),

Calculus of smooth functions between convenient vector spaces, Aarhus Preprint Series 1984/85 No. 18, retyped in TeX 2004.

Synthetic characterization of reduced algebras JPAA 36 (1985), 273-279.

On 1-form classifiers (with E. Dubuc), Communications in Algebra 12 (1984), 1471-1531.

Ehresmann and the fundamental structures ... from a synthetic viewpoint, (retyped from) commentary in C. Ehresmann's Oeuvres completes et commentees (ed. A.C. Ehresmann), Amiens 1984.

Some problems and results in synthetic functional analysis, in "Category Theoretic Methods in Geometry", Proceedings 1983 (ed. A. Kock), Aarhus Various Publication Series No. 35 (1983), 168-191.

A combinatorial theory of connections, in Mathematical Applications of Category Theory (ed. J.W.Gray), AMS Contemporary Math. Vol. 30 (1983) 132-144.

Differential forms with values in groups, Bull. Austral. Math. Soc. 25 (1982), 357-386.

Synthetic Differential Geometry (First Edition), London Math. Soc. Lecture Notes Series 51 (1981), Cambridge University Press. Here is a link to the Second Edition (2006).

A general algebra/geometry duality, and synthetic scheme theory, Prepublications Math., U. Paris Nord 23 (1981), 33-34.

Formal manifolds and synthetic theory of jet bundles, Cahiers 21 (1980), 227-246.

Forms and integration in synthetic differential geometry (with G.E. Reyes and B. Veit), Aarhus Preprint Series 1979/80 no. 33.

Remarks on the Maurer-Cartan forms , in Rapport de Recherces du Dept. de Math. et de Stat., D.M.S. no. 80-12 (ed. G.E.Reyes), 1980.

Formally real local rings, and infinitesimal stability, in Topos Theoretic in Geometry, Proceedings Aarhus 1978 (ed. A. Kock), Aarhus Various Publication Series 30 (1979) 123-136. Retyped in TeX in the winter 2007-2008.

Linear algebra in a local ringed site, Communications in Algebra 3 (1975), 545-561.

Bilinearity and Cartesian closed monads Math.Scand. 29 (1971), 161-174.

Strong functors and monoidal monads Arch.Math. (Basel) 23 (1972), 113-120.

On double dualization monads, Math. Scand. 27 (1970), 151-165.

Closed categories generated by commutative monads, J. Austral. Math. Soc. 12 (1971), 405-424.

Monads on symmetric monoidal closed categories, Archiv.Math. (Basel) 21 (1970), 1-9.

Continuous Yoneda Representation of a small category , Preprint October 1966.

Provisional - not yet sorted.

Also, here will (later in February or March 2020) appear unpublished items, lectures etc.

The dual fibration in elementary terms We give an elemetary construction of the dual fibration of a fibration. It does not use the non-elementary notion of (pseudo-) functor into the category of categories. (on arXiv 1501.019479

Luminy talk 2010

Proprieta dell' anello generico (notes by B. Veit), Rome 1977.

Fractional exponent functors and categories of differential equations (with Reyes) 1998, unpublished. The category theoretic aspects are largely subsumed in Aspects of Fractional Exponents (TAC article, link above), but the differential equations aspects are treated more deeply.

Uldum talk 1971

(with C.J. Mikkelsen) Topos theoretic factorization of non-standard extensions, in Victoria Symposium on Nonstandard Analysis 1972, SLN 369

In the editorial committee of the journals:

Cahiers de Topologie et Geometrie Differentielle Categoriques
I was in the editorial committee of Beitraege zur Algebra und Geometrie in the period Jan. 1993-Jan. 2013. and
Theory and Applications of Categories in the period .. 2019.

Pictures:

Some pictures of me, (phot. by M. Djordjevic), from a seminar talk given at the Mittag-Leffler Institute in June 2001, are available here , here , or here . In the last of them, Steve Awodey seems to be listening carefully.
Finally, a drawing Cape St. Mary I made while in Atlantic Canada (Nova Scotia) in 1969-1970 (Dalhousie Topos year), where I learned so much.

This page last updated April 15, 2023.
Anders Kock