Orthodox synthetic differential geometry is concerned with microlinear spaces. This paper explains how to develop synthetic differential geometry of microlinear stacks or microlinear categories. As illustrations, we will address the categorical Lie algebra of infinitesimal endofunctors and infinitesimal natural transformations, the theory of differential forms and the categorified Ambrose-Palais-Singer theorem. In this paper, we use the word “stack” as a synonym of “category”.
References
[1]
Kock, A. (2006) Synthetic Differential Geometry. 2nd Edition, Cambridge University Press. https://doi.org/10.1017/cbo9780511550812
[2]
Lavendhomme, R. (1996) Basic Concepts of Synthetic Differential Geometry. Kluwer Academic Publishers.
[3]
Ambrose, W.A., Palais, R.S. and Singer, I.M. (1960) Sprays. Anais da Academia Brasileira de Ciências, 32, 163-178.
[4]
Bungé, M. and Sawyer, P. (1984) On Connections, Geodesics and Sprays in Synthetic Differential Geometry. Cahiers de Topologie et Géométrie Différentielle Catégoriques, 25, 221-258.
[5]
Kock, A. and Lavendhomme, R. (1984) Strong Infinitesimal Linearity, with Applications to Strong Difference and Affine Connections. Cahiers de Topologie et Géométrie Différentielle Catégoriques, 25, 311-324.
[6]
Nishimura, H. (1997) Theory of Microcubes. International Journal of Theoretical Physics, 36, 1099-1131. https://doi.org/10.1007/bf02435803
[7]
Nishimura, H. (1997) General Jacobi Identity Revisited. International Journal of Theoretical Physics, 38, 2163-2174. https://doi.org/10.1023/a:1026670206427
[8]
Nishimura, H. and Osoekawa, T. (2007) General Jacobi Identity Revisited Again. International Journal of Theoretical Physics, 46, 2843-2862. https://doi.org/10.1007/s10773-007-9397-z
