Abstract
The theory of combinatorial differential forms is usually presented in simplicial terms. We present here a cubical version; it depends on the possibility of forming affine combinations of mutual neighbour points in a manifold, in the context of synthetic differential geometry.
Similar content being viewed by others
References
Breen, L., Messing, W.: Combinatorial differential forms. Adv. Math. 164, 203–282 (2001)
Brown, R., Higgins, P.J.: On the algebra of cubes. J. Pure Appl. Algebra 21, 233–260 (1981)
Brown, R., Spencer, C.: Double groupoids and crossed modules. Cahiers Topologie Géom. Différentielle Catég. 17, 343–362 (1976)
Felix, Y., Lavendhomme, R.: On DeRham’s theorem in synthetic differential geometry. J. Pure Appl. Algebra 65, 21–31 (1990)
Grandis, M., Mauri, L.: Cubical sets and their site. Theory Appl. Categ. 11, 186–211 (2003)
Hilton, P.J., Wylie, S.: Homology Theory. University Press, Cambridge (1960)
Kock, A.: Synthetic Differential Geometry (Cambridge U.P. 2006), 2nd edn. University Press, Cambridge (1981)
Kock, A.: Combinatorics of curvature, and the Bianchi identity. Theory Appl. Categ. 2, 69–89 (1996)
Kock, A.: Geometric construction of the Levi-Civita parallelism. Theory Appl. Categ. 4, 195–207 (1998)
Kock, A.: Differential forms as infinitesimal cochains. J. Pure Appl. Algebra 154, 257–264 (2000)
Kock, A.: Connections and path connections in groupoids. Aarhus Math. Inst. 10, (2006, preprint)
Kock, A.: Some matrices with nilpotent entries, and their determinants. arXiv:0612435 [math.RA]
Kock, A.: Group valued differential forms revisited. Aarhus Math. Inst. 1 (2007, preprint)
Kock, A.: Infinitesimal cubical structure, and higher connections. arXiv:0705.4406 [math.CT]
Kock, A., Reyes, G.E., Veit, B.: Forms and integration in synthetic differential geometry. Aarhus Math. 31 (1979/1980, preprint series)
Lavendhomme, R.: Basic Concepts of Synthetic Differential Geometry. Kluwer, Deventer (1996)
Meloni, G.-C., Rogora, E.: Global and infinitesimal observables. In: Borceux, F. (ed.) Categorical Algebra and its Applications, Proceedings, Louvain-la-Neuve 1987. Springer Lecture Notes. vol. 1348, pp. 270–279. Springer, Berlin (1988)
Moerdijk, I., Reyes, G.E.: Models for Smooth Infinitesimal Analysis. Springer, Berlin (1991)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kock, A. Cubical Version of Combinatorial Differential Forms. Appl Categor Struct 18, 165–183 (2010). https://doi.org/10.1007/s10485-008-9143-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-008-9143-6