Tangent categories provide an axiomatic framework for understanding various tangent bundles and differential operations that occur in differential geometry, algebraic geometry, abstract homotopy theory, and computer science. Previous work has shown that one can formulate and prove a wide variety of definitions and results from differential geometry in an arbitrary tangent category, including generalizations of vector fields and their Lie bracket, vector bundles, and connections. In this paper we investigate differential and sector forms in tangent categories. We show that sector forms in any tangent category have a rich structure: they form a symmetric cosimplicial object. This appears to be a new result in differential geometry, even for smooth manifolds. In the category of smooth manifolds, the resulting complex of sector forms has a subcomplex isomorphic to the de Rham complex of differential forms, which may be identified with alternating sector forms. Further, the symmetric cosimplicial structure on sector forms arises naturally through a new equational presentation of symmetric cosimplicial objects, which we develop herein.
Differential categories Tangent categories Differential forms de Rham cohomology Simplicial objects Sector forms
Mathematics Subject Classification
18D99 58A10 58A12 58A32 51K10 55U10 55U15 18G30
This is a preview of subscription content, log in to check access.
Kock, A.: Differential Forms in Synthetic Differential Geometry. Aarhus Preprint Series (28) (1978/79)Google Scholar
Kock, A.: Differential forms as infinitesimal cochains. J. Pure Appl. Algebra 154(1–3), 257–264 (2000). Category theory and its applications (Montreal, QC, 1997)MathSciNetCrossRefGoogle Scholar
Kock, A.: Synthetic differential geometry, volume 333 of London Mathematical Society Lecture Note Series, second edition. Cambridge University Press, Cambridge (2006). http://home.imf.au.dk/kock/sdg99.pdf
Kolář, I., Michor, P.W., Slovák, J.: Natural Operations in Differential Geometry. Springer, Berlin (1993)CrossRefGoogle Scholar
Kriegl, A., Michor, P.W.: The Convenient Setting of Global Analysis, volume 53 of Mathematical Surveys and Monographs, vol. 53. American Mathematical Society, Providence, RI (1997)CrossRefGoogle Scholar
Lafont, Y. Equational reasoning with 2-dimensional diagrams. In: Term rewriting (Font Romeux, 1993), volume 909 of Lecture Notes in Comput. Sci., pp. 170–195. Springer, Berlin (1995)Google Scholar
Mac Lane, S.: Categories for the Working Mathematician. Springer, New York (1971). Graduate Texts in Mathematics, Vol. 5CrossRefGoogle Scholar
Manzonetto, G.: What is a categorical model of the differential and the resource \(\lambda \)-calculi? Math. Struct. Comput. Sci. 22(3), 451–520 (2012)MathSciNetCrossRefGoogle Scholar
Moerdijk, I., Reyes, G.E.: Cohomology theories in synthetic differential geometry. Rev. Colombiana Mat. 20(3–4), 223–276 (1986). Seminar and workshop on category theory and applications (Bogotá, 1983)MathSciNetzbMATHGoogle Scholar
Moerdijk, I., Reyes, G.E.: Models for Smooth Infinitesimal Analysis. Springer, New York (1991)CrossRefGoogle Scholar
Rahula, M., Vašík, P., Voicu, N.: Tangent structures: sector-forms, jets and connections. J. Phys. Conf. Ser. 346, 1–9 (2012)CrossRefGoogle Scholar