Anders Kock

Areas of particular interest:

Recent Publications and Preprints

• Abstract projective lines (Dec. 2009). 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.
• Affine connections, midpoint formation, and point reflection (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) (Partly expository). We show how symmetric affine connections are equivalent to midpoint-formation for second order neighbours.
• Cubical version of combinatorial differential forms (November 2007; minor revisions April 2008. To appear in Applied categorical Structures.) 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.
• 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.
• 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) 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. They deal with what is now often called "KZ-monads". One version (with inadequate diagrams) 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.

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Some old published things, scanned to pdf:
• Monads on symmetric monoidal closed categories, Archiv der Math. 21 (1970), 1-10.
• 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.

• The category aspect of projective space , Aarhus Preprint 1974/75 no. 7, (scanned). How points in a projective space are the objects of a groupoid, where an arrow between any two points is a third point, collinear with the given two; this gives cross ratio as a special case of composition. This theory has been developed further by Diers and Leroy, cf. Cahiers, Vol. 35 (1994).
• 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).
• Local groupoids and etendues , talk at 54 PSSL, Sussex March 1994
• 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.)
• 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 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.)
• A popular lecture on knots , 2002.
• On a theorem of Lauchli concerning proof bundles , 1970 (scanned - 12 MB !).

In the editorial committee of the journals:

• Cahiers de Topologie et Geometrie Differentielle Categoriques
• Beitraege zur Algebra und Geometrie
• Theory and Applications of Categories

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.