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 (Feb. 2019) under reconstruction from my old blue
homepage, - so, some links may be dead, or duplicated, or unexplained.
Also, it is not intended to be complete. This in particular applies to
items prior to the age of TeX, say prior to 1989.
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.
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
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. - Accepted for
publication in Cahiers 2019
Affine combinations in
We show how to form affine combinations of
p-tuples of mutually neighbouring points in an
affine scheme. In Cahiers 58
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.
We explore how the synthetic theory of
metric spaces (Busemann) can coexist with synthetic
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
Huygens' principle - a synthetic account . (On arXiv
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
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
right adjoint between slices of E is given by a polynomial. The slices in
question are taken in a well known fibered sense.
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.
midpoint formation, and point reflection, Theoretical Computer Science 412 (2011), 4770-4777.
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
Press has exclusive copyright (of the final version), so please do
not circulate this preliminary version.
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
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
- 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),
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.
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
Connections and path connections in groupoids
Aarhus Math. Preprint Series 2006 no. 10.
Categorical distribution theory: heat equation (with G.E. Reyes)
Categorical distribution theory: heat equation (with G.E. Reyes)
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.
and nilpotent real numbers, Bulletin of Symbolic Logic 9 (2003),
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
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.
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
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.
curvature, and the Bianchi identity,
TAC, Vol 2 (1996), 69-89.
Glueing analysis for
(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
Relatively Boolean an de Morgan toposes and locales, (with G.E. Reyes), Cahiers 35 (1994),
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
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
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.
and left exactness of Kan Extensions
Aarhus Preprint 1989/90 no. 9, Retyped in TeX in the fall of 2003.
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
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
spaces embed into the Cahiers topos Cahiers de topologie et
geometrie differentielle categoriques 27 (1986), 3-17.
addenda to Convenient vector
spaces embed .. (with G.E. Reyes), Cahiers de topologie et
geometrie differentielle categoriques 28 (1987), 99-110.
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
Austral. Math. Soc. 34 (1986), 395-410
Synthetic reasoning in
differential geometry, Revista Colombiana de Mat. 20 (1986), 129-146.
relating to principal fibre bundles, JPAA 39 (1986), 141-151.
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.
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
Ehresmann's Oeuvres completes et commentees (ed. A.C. Ehresmann), Amiens
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
algebra/geometry duality, and synthetic scheme theory,
Prepublications Math., U. Paris Nord 23 (1981),
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
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.
generated by commutative monads, J. Austral. Math. Soc. 12 (1971),
Monads on symmetric
monoidal closed categories, Archiv.Math. (Basel) 21 (1970), 1-9.
Representation of a small category , Preprint October 1966.
This page last updated March 4, 2019.
Provisional - not yet sorted.
Also, here will (later in February or March 2019) appear unpublished items, lectures etc.
fibration in elementary terms
We give an elemetary construction of the
of a fibration. It does
not use the non-elementary notion of (pseudo-) functor into the category
of categories. (on arXiv 1501.019479
anello generico (notes by B. Veit), Rome 1977.
functors and categories of differential equations (with
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.
In the editorial committee of the journals:
Some pictures of me, (phot. by M. Djordjevic), from a seminar talk given
at the Mittag-Leffler Institute in June 2001, are available
, 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.