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.

2020 +

Slides for my talk at the CT 2023 in Louvain-La-Neuve in memory of
Bill Lawvere, 1937-2023.

Obituary for F. William Lawvere,
1937-2023. Published in European Math. Soc. Magazine 128, June 2023

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.)

Heron's formula, and
volume forms
Heron's formula from antiquity is used to relate volume form and
infinitesimal square-volume of certain infinitesimal simplices in a
Riemannian manifold. (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

2010 - 2019

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.

** 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
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.

Fibrations as
Eilenberg-Moore algebras. No claim of
originality; the account avoids
any mention of category-valued functors and pseudofunctors.

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.

Extensive quantities
and monads
Manuscript for an expository talk given at Krakow (DGMP) in 2011.

Monads and extensive quantities
and monads Preliminaries and motivation for my 2012
article in TAC on commutative monads as a
theory of distributions.

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

2000 - 2009

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.

Some matrices with nilpotent elements and their
determinants This contains the basic linear algebra used for the
infinitesimal simplices which classify differential forms (as expounded in
Syntetic Geometry of Manifolds).

**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 Any pregroupoid may
be canonically realized as the set of arrows from A to B, where A and
B are two disjoint subsets of the set of objects of the groupoid.

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

Classifying surjective equivalences How to construct a groupoid
out of a cocycle for a principal bundle. (2004).

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.
** Luminy
talk 2010**

1990 -1999

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.

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.

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

1980 - 1989

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.

1965 - 1979

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.

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

Universal
projective geometry via topos theory, JPAA 9 (1976) 1-24

Linear algebra and
projective geometry in the Zariski topos, Aarhus Preprint Series 1974/75
No. 4

The category aspect
of projective space (Aarhus Preprint Series 1974-75 No. 7.)
The set of 1-dimensional linear subspaces of a 2-dimenisonal vector space
form in a natural way the set of objects of a groupoid; an arrow from one
object A to another one B is given by parallel projection in the direction of a
third objcet C. This provides the groupoid with an interesting interplay between
objects and arrows.

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

(with C.J.
Mikkelsen) Topos
theoretic factorization of non-standard extensions, in Victoria Symposium
on Nonstandard Analysis 1972, SLN 369.
Uldum
talk 1971

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

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

Some Topos Thoretic Concepts of Finiteness (joint with
P.Lecouturier and C.J.
Mikkelsen, in Model Theory and Topoi, Springer Lecture Notes in Math. 445
(1974). 209-283

Elementary Toposes (joint with
G.C. Wraith), Aarhus Math. Lecture Notes Series No. 30
(1971)

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

Strong functors and
monoidal monads Arch.Math. (Basel) 23 (1972), 113-120.
Monads on symmetric
monoidal closed categories, Archiv.Math. (Basel) 21 (1970), 1-9.

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

### In the editorial committee of the journals:

### 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 Dec. 1, 2023.
Anders Kock