Homepage of

Anders Kock

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.
  • Huygens' Principle Manuscript for invited talk at CT 2018, Ponta Delgada (Azores), Portugal.

  • Publications and Preprints

  • Huygens' Principle - a Synthetic Account A simplified version of part of the paper Metric Spaces and SDG (see next item), with emphasis on the contact-element (wave-front) viewpoint. (Preprint on arXiv 1804.05649.)
  • 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. (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)
  • 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. (Revised and slightly augmented Sep. 2017).
  • Affine 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).
  • Bundle functors and fibrations 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.
  • 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 representables.
  • Projective 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.
  • Closed limit 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 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, 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 .
  • Commutative 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 monads. .
  • Calculus of extensive quantities (May 2011). This is a companion to Monads and extensive quantities (see below) dealing in particular with one-variable differential calculus
  • Monads and extensive quantities (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 monad.
  • Theory of 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.
  • Geometric algebra 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.
  • Abstract projective lines 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 MB). Cambridge University Press has exclusive copyright (of the final version), so please do not circulate this preliminary version in printed form.
  • Affine 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.
  • Affine 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.
  • Cubical 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.
  • Infinitesimal 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.
  • Group valued 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!
  • Connections 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.

  • Synthetic Differential Geometry . 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.
  • Commutation structures , November 2005. We prove a formal result which implies that the tensorial strength of a strong endofunctor with a strong right adjoint is invertible.
  • Principal bundles, 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 of pregroupoid.
  • (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), 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). We study complete solutions on such "exponential" vector fields. These solutions are, from a more standard viewpoint, solutions of certain partial differential equations.
  • Classifying surjective equivalences , 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.
  • 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.
  • A geometric theory of harmonic and semi-conformal maps . This is an improvement and extension of some aspects of the "Laplace" theory 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, Theory 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 context.
  • First neighbourhood of the diagonal, and geometric distributions, Universitatis Iagellonicae Acta Mathematica 41 (2003), 307-318 , 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.
  • (with G.E. Reyes) Some differential equations in SDG, here or arXiv:math/0104164[mathCT]. Most of this is subsumed in our two papers on Wave Equation, and on Heat Equation
  • Algebra of Principal Fibre Bundles, and connections, rejected by TAC 2002, but available here ; 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 pregroupoid ).
  • The stack quotient of a groupoid,  Cahiers de Top. et Geom. Diff. Categorique 44 (2003), 85-104. Preprint version (March 2002) available  here. A picture of the author, presenting a relevant diagram from this theory, is available here (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. Preprint here .
  • 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, arXiv:math.CT/0006008 . 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 sides.
  • Differential Forms as Infinitesimal Cochains. Journ. Pure Appl. Algebra 154 (2000), 257-264 (Preprint 1997 available here . ) 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.
  • The osculating plane of a space curve - synthetic formulations , Rend.Circ.Mat. Palermo II Vol. 64 (2000), 67-79. Preprint version here. 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 , T 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 here . 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 (JPAA 104 (1995) 41-59) is available here. The version from Feb. 1992 is the most algebraic of the versions; it appeared as an Aarhus Preprint, and appears recompiled here.
  • 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 foundations.
  • (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 , Bull. 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 , Bull. Austral. Math. Soc. 34 (1986), 395-410 .
  • On the Integration Theorem for Lie Groupoids, Czechoslovak Math. J. 39 (114), 1989.
  • Linear algebra in a local ringed site, Communications in Algebra 3 (1975), 545-561.
  • Monads on symmetric monoidal closed categories, Archiv 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, Archiv der Math. 23 (1972), 113-120.
  • On the synthetic theory of vector fields, Aarhus Preprint Oct. 1978.
  • A combinatorial theory of connections, in Mathematical 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 publications/prepublications:

  • Differential Geometry Without Real Numbers (1979-80), scanned version of lecture notes for a graduate course. For technical reasons, it the scanned version consists of three files, plus a 2018 preface; so you need to download four files, Preface 2018, plus Part 1, Part2, and Part3.
  • Introduction 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).
  • Trace of 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 (1980)
  • Extension Theory for Local Groupoids. October 1997
  • The Maximal Atlas of a Foliation , (Lecture at 62. PSSL, Utrecht Oct. 1996).
  • Natural bundles over smooth etendues , Sept. 1995, (Lecture at CT95, Halifax July 1995).
  • The Constructive Lift Monad , BRICS Report Series, RS-95-20 (March 1995).
  • Postulated colimits 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.
  • Infinitesimal Deformations of Complete Vector Fields are Complete , Aarhus Preprint 1985/86 no. 23, Retyped in TeX in Jan. 2006.
  • On 1-form classifiers, 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 winter 2006-2007.
  • A general 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 viewpoint Commentary in Vol I-1, I-2 of Ehresmann Oeuvres Completes, ed. A.C. Ehresmann, Amiens 1984, p. 549-554.
  • (with G. Reyes and B. Veit) Forms and integration in synthetic differential geometry , Aarhus Preprint Series 1979/80 No. 31.
  • (notes by B. Veit) Proprieta dell anello generico , Aarhus Preprint 1977
  • 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.
  • 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 fall 2007.
  • Continuous Yoneda representation of a small category . Aarhus University Preprint (un-numbered) 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 on knots , 2002.

  • In the editorial committee of the journals:


    This page last updated Sept. 26, 2018
    Anders Kock