Hostname: page-component-54dcc4c588-ff9ft Total loading time: 0 Render date: 2025-10-06T06:11:06.623Z Has data issue: false hasContentIssue false
  • English
  • Français

Why the Kock-Lawvere axiom holds, and also doesn’t, in the classifier of integral rigs

Published online by Cambridge University Press:  06 October 2025

Matías Menni Open the ORCID record for Matías Menni [Opens in a new window]
Matías Menni*
Affiliation:
Conicet - Centro de Matemática de La Plata, La Plata, Argentina
*
Email: matias.menni@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

The Kock-Lawvere axiom has two formulations that are equivalent in the usual models of Synthetic Differential Geometry. We show that, in the classifier of integral rigs, and some of its pre-cohesive subtoposes, the generic model satisfies one version of the axiom but not the other.

Keywords

Rig geometryaxiomatic cohesionsynthetic differential geometryTopos theory

Information

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 35 , 2025 , e25
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Atiyah, M. F. and Macdonald, I. G. (1969). Introduction to Commutative Algebra, Addison-Wesley Publishing Co. Reading, Mass.-London-Don Mills, Ont.Google Scholar
Castiglioni, J. L., Menni, M. and Zuluaga Botero, W. J. (2016). A representation theorem for integral rigs and its applications to residuated lattices. Journal of Pure and Applied Algebra 220 (10) 3533–3566. https://doi.org/10.1016/j.jpaa.2016.04.014.CrossRefGoogle Scholar
Cockett, J. R. B. and Cruttwell, G. S. H. (2014). Differential structure, tangent structure, and SDG. Applied Categorical Structures 22 (2) 331–417.10.1007/s10485-013-9312-0CrossRefGoogle Scholar
Johnstone, P. T. (2002). Sketches of an Elephant: A Topos Theory Compendium, Vol. 43-44 of Oxford Logic Guides, New York, The Clarendon Press Oxford University Press.Google Scholar
Kock, A. (2006). Synthetic Differential Geometry, 2nd edn, Cambridge, Cambridge University Press.10.1017/CBO9780511550812CrossRefGoogle Scholar
Kock, A. (2017). Affine combinations in affine schemes. Cahiers de Topologie et Géométrie Différentielle Catégoriques 58 (2) 115–130.Google Scholar
Lawvere, F. W. (2007). Axiomatic cohesion. Theory and Applications of Categories 19 41–49.10.70930/tac/rwez60o2CrossRefGoogle Scholar
Lawvere, F. W. (2008). Core varieties, extensivity, and rig geometry. Theory and Applications of Categories 20 (14) 497–503.10.70930/tac/ag8jrxkbCrossRefGoogle Scholar
Lawvere, F. W. (2011). Euler’s continuum functorially vindicated. In: Logic, Mathematics, Philosophy: Vintage Enthusiasms’, Vol. 75 of The Western Ontario Series in Philosophy of Science, Springer Science+Bussiness Media B. V. 249–254.10.1007/978-94-007-0214-1_13CrossRefGoogle Scholar
Marmolejo, F. and Menni, M. (2019). Level $epsilon$ . Cahiers de Topologie et Géométrie Différentielle Catégoriques 60 (4) 450–477.Google Scholar
Menni, M. (2021). A basis theorem for 2-rigs and rig geometry. Cahiers de Topologie et Géométrie Différentielle Catégoriques 62 (4) 451–490.Google Scholar
Menni, M. (2024). Bi-directional models of radically synthetic differential geometry. Theory and Applications of Categories 40 413–429.10.70930/tac/xnycc7b9CrossRefGoogle Scholar
Schanuel, S. H. (1991). Negative sets have Euler characteristic and dimension. In: Category Theory, Proc. Int. Conf., Como/Italy 1990, Lect. Notes Math. Vol. 1488, Springer, 379–385.10.1007/BFb0084232CrossRefGoogle Scholar
Spada, L. and St. John, G. (2025). Weil 2-rigs. Theory and Applications of Categories 43 (12) 382–402.10.70930/tac/qgtafibnCrossRefGoogle Scholar