Affine Connections, and Midpoint Formation

  • Anders Kock
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5810)


It is a striking fact that differential calculus exists not only in analysis (based on the real numbers \(\mathbb R\)), but also in algebraic geometry, where no limit processes are available. In algebraic geometry, one rather uses the idea of nilpotent elements in the “affine line” R; they act as infinitesimals. (Recall that an element x in a ring R is called nilpotent if x k  = 0 for suitable non-negative integer k.)

Synthetic differential geometry (SDG) is an axiomatic theory, based on such nilpotent infinitesimals. It can be proved, via topos theory, that the axiomatics covers both the differential-geometric notions of algebraic geometry and those of calculus.


Algebraic Geometry Category Theory Geometric Construction Nilpotent Element Bijective Correspondence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Breen, L., Messing, W.: Combinatorial differential forms. Advances in Math. 164, 203–282 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Kock, A.: Synthetic Differential Geometry. London Math. Soc. Lecture Notes Series, vol. 51. Cambridge University Press, Cambridge (1981)*(Second edition, London Math. Soc. Lecture Notes Series, vol. 333. Cambridge University Press, Cambridge (2006))Google Scholar
  3. 3.
    Kock, A.: A combinatorial theory of connections. In: Gray, J. (ed.) Mathematical Applications of Category Theory, Proceedings 1983. AMS Contemporary Math., vol. 30, pp. 132–144 (1984)Google Scholar
  4. 4.
    Kock, A.: Introduction to Synthetic Differential Geometry, and a Synthetic Theory of Dislocations. In: Lawvere, F.W., Schanuel, S. (eds.) Categories in Continuum Physics, Proceedings Buffalo 1982. Springer Lecture Notes, vol. 1174. Springer, Heidelberg (1986)Google Scholar
  5. 5.
    Kock, A.: Geometric construction of the Levi-Civita parallelism. Theory and Applications of Categories 4, 195–207 (1998)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Kock, A.: Differential forms as infinitesimal cochains. Journ. Pure Appl. Alg. 154, 257–264 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kock, A.: Synthetic Geometry of Manifolds. Cambridge Tracts in Mathematics 180 (2009)Google Scholar
  8. 8.
    Noll, W.: Materially uniform simple bodies with inhomogeneities. Arch. Rat. Mech. Anal. 27, 1–32 (1967)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Anders Kock
    • 1
  1. 1.Department of Mathematical SciencesUniversity of AarhusAarhus CDenmark

Personalised recommendations