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)
topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
Given a given intuitionistic logic framework, such as a type theory, the logical topology (Penon 81) on a given object (type) $X$ is that whose open subsets are the sets $U$ for which any $X - \{x\}$ and $U$ cover $X$. This implies that an open subset contains for every point $x$ also the collection of points that are indistinguishable from it in classical logic, hence that are not not equal to $x$.
The notion of Penon open is closely related to that of Zariski open. Consider the affine line $\mathbb{A} : \text{fpRing} \to \text{Set}$ as the forgetful functor from finitely presented rings to sets. The (category theorists’) big Zariski topos is the largest subtopos of this topos of functors in which $\mathbb{A}$ is a local ring. A ring is local if for all $y$, either $y$ or $1- y$ is invertible; equivalently, we may ask that either $y$ or $x - y$ is invertible for a given invertible element $x$, by homogeneity. In this case, the locality of $\mathbb{A}$ is equivalent to the subset $\mathbb{G}_m \hookrightarrow \mathbb{A}$ of invertible elements being logically open.
The concept was introduced in
Review in the context of cohesive toposes, modal type theory and cohesive homotopy type theory includes
Last revised on July 26, 2022 at 14:26:42. See the history of this page for a list of all contributions to it.