analysis (differential/integral calculus, functional analysis, topology)
metric space, normed vector space
open ball, open subset, neighbourhood
convergence, limit of a sequence
compactness, sequential compactness
continuous metric space valued function on compact metric space is uniformly continuous
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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
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
symmetric monoidal (∞,1)-category of spectra
Since every ordered local ring $R$ has characteristic zero, the positive integers $\mathbb{Z}_+$ are a subset of $R$, with injection $i:\mathbb{Z}_+ \hookrightarrow R$. An Archimedean ordered local ring is an ordered local ring which satisfies the archimedean property: for all elements $a \in R$ and $b \in R$, if $0 \lt a$ and $0 \lt b$, then there exists a positive integer $n \in \mathbb{Z}_+$ such that such that $a \lt i(n) \cdot b$.
Unlike Archimedean ordered fields, which require arithmetic Heyting pretoposes, Archimedean ordered local rings are definable in any arithmetic pretopos.
Archimedean ordered local $\mathbb{R}$-algebras are important for modeling notions of infinitesimals. These include the dual numbers, which represent nilsquare infinitesimals and are used to synthetically define differentiable functions in the real numbers, Archimedean ordered Weil $\mathbb{R}$-algebras, which represent nilpotent infintiesimals and are used to synthetically define smooth functions in the real numbers, as well as formal power series on $\mathbb{R}$, which represent infinitesimals which are not nilpotent and are used to synthetically define analytic functions in the real numbers.
Last revised on January 13, 2023 at 08:01:16. See the history of this page for a list of all contributions to it.