At the Department of Mathematical Sciences
of Aarhus University
Office: Building 1535, 2.29
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.
A chronological list (.pdf format) of my publications and prepublications,
as of Feb. 2010, is available here . A partial, but commented
and clickable, list in .html format appears below.
At the bottom of the page, you will find retypings in TeX (.pdf
format) or scannings of
some older publications.
Publications and Preprints
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. (The present version June 2017, to appear in
Theory and Applications of Categories, contains some
corrections and reorganizations, compared to the arXiv version Nov.1
2016, as well as to the versions 2, 3 and 4)
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. (Revised and slightly augmented Sep.
combinations in affine schemes
We show how to form affine combinations of
p-tuples of mutually neighbouring points in an
affine scheme. Appeared in Cahiers de Top. et Geom. Diff 58
(2017), 115-130, (and on arXiv 1508.04322).
functors and fibrations
We give an account of bundle-functors and
star-bundle-functors (known from differential geometry)
in terms of fibered categories.
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
Duality for generic
algebras, Cahiers de Top. et Geom. Diff. Categorique 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, No. 13 (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.
formula for higher derivatives
We give a simple construction for a parametrized family of
linear combination of n+1 Dirac
distributions on the line, which in the limit is the distribution taking
n-fold derivative at 0.
(joint with J.
Kock) Local fibered right adjoints are polynomial
, Math. Structures
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, No.4 (2012), 97-131. It summarizes and simplifies some of the contents of the two
following items. The notion of commutative monad is in the sense of
my 1970-1972 papers, see below .
monads, distributions, and differential categories
Talk for the Cambridge Category Seminar, Nov. 20, 2012.
Besides a summary of the work on commutative monads, the talk
indicates how differential categories (in the sense of Blute, Cockett,
Seely) grow out of suitable commutative
(May 2011). This is a companion to Monads and extensive quantities (see
dealing in particular with one-variable differential calculus
(March 2011). If T is a commutative monad on a cartesian closed category,
it gives rise to a notion of extensive quantities, in the sense of
Lawvere, encoded by an integration process, or equivalently, by a
monad comparison with a Schwartz type distribution-of-compact-support
characteristics for first order partial differential equations
(November 2010). A conspectus of Lie's theory of the caracteristic stripes
of a first order PDE, - using the method of SDG.
of projective lines
(March 2010). 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.
Cahiers de Top.
et Geom. Diff. Categorique 51 (2010), 224-240. 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.
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 in printed form.
connections, midpoint formation, and point reflection
Theoretical Computer Science 2010 , (see also
preprint version, Jan.
2010). This is an expanded version
of "Affine connections, and midpoint formation" (see next item),
with more proofs included, and also with
description of an (infinitesimal) version of space
with point reflection (geodesic symmetry), equivalent to the two
other types of structure.
connections, and midpoint formation
(June 2009). Appeared in Discrete
Geometry for Computer Imagery, Proceedings of the
15th IAPR Conference, Montreal 2009, Springer LNCS
5810. We show how symmetric affine
connections are equivalent to midpoint-formation
for second order neighbours.
version of combinatorial differential forms
(November 2007; Appl.Categor Struct
(2010) 18:165-183) We present a cubical (rather than
simplicial) version of combinatorial differential
forms. This version is suited for being
generalized into a theory of higher connections,
as sketched in the item below.
cubical structure, and higher connections
, May 2007 (also at arXiv: 0705:4406 [math.CT]). We formulate some theory of higher connections
with values in cubical groupoids in terms of
morphisms of cubical complexes. Any manifold gives, via its first
neighbourhood of the diagonal, rise to a cubical complex consisting of
infinitesimal parallelepipeda. Slides for my talk given at CT 2007
in Carvoeiro are available here.
differential forms revisited
, Aarhus Preprint 2007, No. 1, February 2007. We study the relationship between
combinatorial group valued differential forms and classical differential forms
with values in the corresponding Lie algebra. In particular, we compare
simplicial coboundary and exterior derivative. The results represent
strengthening of results I obtained in 1982.
Some matrices with
nilpotent entries, and their determinants
, December 2006. We study algebraic properties om 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!
and path connections in groupoids
, Aarhus Preprint 2006, No. 10, September 2006. We describe how holonomy (integration) and
a certain differentiation process
establish a bijective correspondence between
connections and path connections in groupoids.
. This book from 1981 has long been out of print. Now a Second
Edition has been published by Cambridge University
Press, London Math. Society Lecture NotesSeries No 333 (June 2006). It contains a retyping in TeX of the First Edition,
and is also supplied with some short updating notes, and an updated
bibliography. The present link is to the
galley proofs of the TeX version. Cambridge University Press has exclusive
copyright to the new version in its printed form, so please do not circulate this
file in printed form.
, November 2005. We prove a formal result which implies that
the tensorial strength of
a strong endofunctor with a strong right adjoint is invertible.
groupoids, and connections
, December 2005. 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
(with G.E. Reyes)
Categorical distribution theory; heat equation
, November 2004. (Appeared in Cahiers de Top. et Geom. Diff.
Categorique 47 (2006), 2-28 with title: Distributions and heat equation.) - We describe how distribution theory
(even with non-compact support) fits into the context of topos models for
Synthetic Differential Geometry. In particular, we deal with the heat equation.
(with G.E. Reyes)
Ordinary differential equations and their exponentials
, September 2004; 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).
We study complete solutions on such "exponential" vector fields. These solutions
are, from a more standard viewpoint, solutions of certain partial differential
, Jan. 2004. For a given groupoid, and given quotient of its
set of objects, we describe the category of equivalences, covering the given
quotient map, in terms of a category of cocycles.
- 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.
A geometric theory of harmonic and semi-conformal maps
. This is an improvement and extension of some aspects of the
begun in my 2001 paper. Presented at the 5 Krynica Conference on Geometry
and Topology of Manifolds, April-May 2003. Appeared in
Central European Journal of Mathematics 2(5) 2004 708-724
Pregroupoids and their enveloping groupoids
. We prove that the forgetful functor from groupoids to pregroupoids,
or equivalently, to a certain category of torsors, has a left adjoint, with
monic unit for the adjunction. This provides a tool for "multiplicative calculations"
in pregroupoids or torsors.
(with G.E. Reyes),
Some calculus with extensive quantities: wave equation,
and Applications of Categories, Vol 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. Our theory applies also in, say, complex-analytic
First neighbourhood of the diagonal, and geometric distributions,
Universitatis Iagellonicae Acta Mathematica 41 (2003),
, or math.DG/0206065
,. This contains a synthetic version of the Ambrose-Singer theorem about
holonomy of connections in principal fibre bundles.
Infinitesimal aspects of the Laplace operator,
Theory and Applications of Categories, Vol 9. (2001), No. 1
It describes a neighbourhood of the diagonal in any Riemannian manifold,
much smaller than the second order neighbourhood, but large enough to carry
information about Laplacian of functions, and of conformality.
Algebra of Principal Fibre Bundles, and connections,
rejected by TAC 2002, but available
; preliminary version at Xiv:math.CT/0005125. This is the "cross-roads", where
my work on "multiplicative" curvature theory
meets my work on fibre bundles from the 1980's (via the notion of
enveloping groupoid of a
The stack quotient of a groupoid, Cahiers de Top.
et Geom. Diff. Categorique 44 (2003), 85-104. Preprint version (March 2002)
A picture of the author, presenting a relevant diagram from this theory, is
(phot: M. Djordjevic, at Mittag-Leffler Institute Stockholm 2001).
Differential calculus, and nilpotent real numbers
, paper presented at "Meeting on Existence in Mathematics", University of
Roskilde, Nov. 24-25 2000 and at the meeting "New programs and open problems
in the foundation of mathematics and of its applications", Paris, November
13 and 14, 2000. Bull. Symb. Logic 9 (2003), 225-230.
Characterization of stacks of principal fibre bundles
Inst itut Mittag-Leffler report 27, 2000/2001
(May 2001), or here.
"Principal fibre bundles" here means torsors over a groupoid.
Volume form as volume of infinitesimal simplices,
. To deduce the volume form out of a Riemannian metric, we use an infinitesimal
form of Heron's Theorem about area of a triangle in terms of lengths of its
Differential Forms as Infinitesimal Cochains. Journ.
Pure Appl. Algebra 154 (2000), 257-264 (Preprint 1997 available
) 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
The osculating plane of a space curve - synthetic formulations
, Rend.Circ.Mat. Palermo II Vol. 64 (2000), 67-79. Preprint version
This proves a well known result in of differential geometry by purely synthetic
means, meaning that no coordinatization of any kind appears.
(with G.E.Reyes), Aspects of Fractional Exponent Functors
heory and Applications
of Categories, 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".
(with G.E.Reyes), Fractional Exponent Functors and Categories
of Differential Equations
, here, November 1998.
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.
(with G.E. Reyes) A note on frame distributions, Cahiers
de Top. et Geom. Diff. Cat 40 (1999), 127-140. Preprint version
. 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.
(with Till Plewe) Glueing Analysis for Complemented Subtoposes
, Theory and Applications
of Categories, Vol. 2,(1996), No.9
. This gives a glueing construction of a topos out of a locally closed subtopos
and its complement
Combinatorics of Curvature, and the Bianchi Identity,
Theory and Applications of Categories, Vol. 2 (1996), No. 7
Using the synthetic method of infinitesimal simplices, for connections and
differential forms, it appears that the Bianchi identity for curvature of
a connection can be deduced from the combinatorics of the boundary 1-skeleton
of a tetrahedron. A machinery (log and exp) is established for a systematic
comparison between the simplicial combinatorics, on the one hand, and the
standard "linear" calculations, on the other.
(with I. Moerdijk) Spaces with local equivalence relations,
and their monodromy, Topology and its Applications 72 (1996), 47-78.
We elaborate on a suggestion of Grothendieck, and study invariant sheaves
for a local equivalence relation on a space; the topos thus arising is a
kind of quotient for the relation.
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, available (scanned)
here. They deal with what is now often called "KZ-monads". The
published version is available
here (but with
. The version from Feb. 1992 is the most algebraic of the versions; it appeared as an Aarhus
Preprint, and appears recompiled
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
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.
Recompiled version available here
Kategori-teori, og matematikkens grundlag, IMFUFA Roskilde 1998.
Essay (in Danish) on
(with J. Schmidt) A
coherent theory of sites ,
Bulletin de la Soc. Math. de Belgique (Serie A), 41 (1989), 321-331.
We describe in coherent (= finitary geometric) language a notion of site.
Generalized fibre bundles ,
in Categorical Algebra and
its Applications, Proceedings, Louvain-La-Neuve 1987 (ed. F. Borceux)
Synthetic reasoning in differential geometry ,
Revista Colombiana de
Matematicas, Vol. 20 (1986), 129-146.
Convenient vector spaces embed into the Cahiers topos ,
Cahiers de topologie et
geometrie differentielle categoriques 27 (1986), 3-17.
(with G.E. Reyes) Corrigendum and addenda to Convenient vector
spaces embed ..
,Cahiers de topologie et
geometrie differentielle categoriques 28 (1987), 99-110.
Some old published things, scanned to pdf:
Differential forms with values in groups ,
Austral. Math. Soc. 25 (1982), 357-386 .
Synthetic characterization of reduced algebras ,
Journ. Pure Appl. Alg. 36
(1985), 273-279 .
Lie group valued integration in well adapted toposes ,
Austral. Math. Soc. 34 (1986), 395-410 .
On the Integration Theorem for Lie Groupoids,
Czechoslovak Math. J. 39
Monads on symmetric monoidal closed categories,
der Math. 21 (1970), 1-10.
Closed categories generated by commutative monads,
J.Australian Math.Soc. 12 (1971), 405-424..
On double dualization monads,
Math. Scand 27 (1970), 151-165.
Bilinearity and Cartesian closed monads,
Math. Scand 29 (1971), 161-174.
Strong functors and monoidal monads,
der Math. 23 (1972), 113-120.
A combinatorial theory of connections,
Applications of Category Theory (ed. J.W.Gray), AMS Contemporary Math.
Vol. 30 (1983) 132-144
. (The link gives the preprint version)
Unpublished items, and retypings of older
to Functorial Semantics
Mimeographed notes for my contribution at Bertrand Russell Memorial Logic
Conference, Uldum (Denmark) 1971. (scanned - almost 20 MB !) The second
half of this gives an analysis in topos theoretic terms of the notion of
internal subset, in terms of a first order logic preserving endo-functor
on the category of sets. This theory was later devloped further by Chr.
Juul Mikkelsen and myself, in a paper for the Victoria symposium on
Nonstandard analysis, Univ. of Victoria 1972, see below.
(with Chr. Juul
Mikkelsen) Topos Theoretic
Factorization of Non-standard Extensions
In Victoria Symposium on
Nonstandard Analysis, Univ. of Victoria 1972, p. 122-143, Springer
Lecture Notes in Math. 369 (1974).
categories, and a universal property of the representation ring of a
finite group. February 1979. The universal property of the latter is that it is the
trace object of the category of complex representations of the group.
These things are studied further in a paper by H. Lindner, in the
Proceedings of the Arnsberg Conference, Hagen Fernuni. Seminarberichte 7
Extension Theory for
The Maximal Atlas
of a Foliation
, (Lecture at 62. PSSL, Utrecht Oct. 1996).
over smooth etendues
, Sept. 1995, (Lecture at CT95, Halifax July 1995).
, BRICS Report Series, RS-95-20 (March 1995).
and left exactness of Kan Extensions
, Aarhus Preprint 1989/90 no. 9, Retyped in TeX in the fall of 2003.
Calculus of smooth functions
between convenient vector spaces
, Aarhus Preprint 1984/85 no. 18, Retyped in TeX in the spring of 2004.
of Complete Vector Fields are Complete
, Aarhus Preprint 1985/86 no. 23, Retyped in TeX in Jan. 2006.
joint with E. Dubuc, from Communications in Algebra 12 (1984), 1471-1531
Some problems and
results in synthetic functional analysis
, in Category Theoretic in Geometry, Proceedings Aarhus 1983 (ed. A. Kock),
Aarhus Various Publication Series 35 (1983) 168-191, Retyped in TeX in the
algebra/geometry duality, and synthetic scheme theory
Prebublications Math., U. Paris Nord 23 (1981), 33-34. Announcing
a result that 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. (Retyped Jan. 2009.)
(with G. Reyes and
B. Veit) Forms and
integration in synthetic differential geometry
, Aarhus Preprint Series 1979/80 No. 31.
Ehresmann and the
fundamental structures of differential geometry seen from a synthetic
Commentary in Vol I-1, I-2 of Ehresmann Oeuvres Completes, ed. A.C.
Ehresmann, Amiens 1984, p. 549-554.
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
On a theorem of
Lauchli concerning proof bundles
, 1970 (poorly scanned - 12 MB !).
On the codensity
monad of a functor . Notices of the AMS
14, 645-49 (April 1967). The codensity monad
of a functor considered here. A result about a limit-preserving property
of a resulting functor is proved. (This notice is a report on the result
in the autors' Continuous Yoneda representation of a small category,
mimeographed Aarhus 1966 (as referenced in Linton's An outline of
functorial semantics, Seminar on Triples and Categorical Homology Theory,
SLN 80 (1969).) See item below.) - Retyped in TeX in the
representation of a small category . Aarhus University Preprint
October 1966. It is proved that a small category embeds in a bi-continuous
way into the category of algebras for the codensity monad of
its Yoneda embedding. (Scanned - 8 MB.)
Limit Monads in
Categories . Ph.D thesis, University of Chicago Sept.\ 1967
A syntactic construction of free cocompletions (for various
kinds of colimits) of categories. Some of the contents was later
crystallized into my papers on Monads for which structures are adjoint to
units. It also by set theoretic methods (ordinal numbers) succeeds in
having the colimit formation strictly associative.
Limit Monads in
Categories . Aarhus University Preprint
1967/68 No. 6,
December 1967. Abridged version of my Ph.D.
thesis (University of Chicago, Sept. 1967). Some of the contents was later
crystallized into my papers on Monads for which structures are adjoint
to units. (KZ.monads)
A popular lecture
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.
This page last updated Feb. 19, 2018