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 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

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

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