synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Euler-Lagrange equation, de Donder-Weyl formalism?,
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory
The notion of differential categories (Blute, Cocket & Seely 2006 is meant to provide categorical semantics for differential linear logic (Ehrhardt & Regnier 2009) which in turn is meant to be a syntactic proof-theoretic approach to differential calculus.
The notion is due to:
R. F. Blute, J. R. B. Cockett, and R. A. G. Seely: Differential categories. Math. Struct. Comput. Sci. 16(06), 1049–1083 (2006) (doi:10.1017/S0960129506005676)
Richard Blute, Robin Cockett, Jean-Simon Lemay and Robert Seely, Differential categories revisited, Appl. Categ. Struct. 28, 171-235 (2020). (arXiv:1806.04804, doi:10.1007/s10485-019-09572-y)
following discussion of differential linear logic in:
Thomas Ehrhard, Laurent Regnier: The differential lambda-calculus, Theor. Comput. Sci. 309 1 (2003) 1–41 (doi:10.1016/S0304-3975(03)00392-X)
Thomas Ehrhard, Laurent Regnier: Differential interaction nets, Theor. Comput. Sci. 364 2 (2006) 166–195 (doi:10.1016/j.tcs.2006.08.003)
Last revised on July 12, 2021 at 10:23:30. See the history of this page for a list of all contributions to it.