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)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(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)
For a topological space satisfying the $R_0$ regularity condition (which states that the specialisation preorder is symmetric, hence an equivalence relation), see separation axioms.
A symmetric space is a specially nice homogeneous space, characterized by the property that for each point there is a symmetry fixing that point and acting as $-1$ on its tangent space. An example would be the sphere, the Euclidean plane, or the hyperbolic plane.
A symmetric space is classically defined to be a quotient manifold of the form $G/H$, where $G$ is a Lie group and the subgroup $H$ is the set of fixed points of some involution $\sigma : G \to G$, that is, a smooth homomorphism with $\sigma^2 = 1_G$. Using the involution, every point $a \in G/H$ gives rise to a smooth function
fixing the point $a$ and acting as $-1$ on the tangent space of $a$. This operations satisfies the laws of an involutory quandle.
More precisely, a symmetric pair is a pair $(G,H)$ where $G$ is a Lie group and the subgroup $H$ is the set of fixed points of some involution $\sigma : G \to G$. Different pairs $(G,H)$, $(G',H')$ can give what is normally considered the same symmetric space $G/H \cong G'/H'$. In other words, not every morphism of symmetric spaces arises from a morphism of symmetric pairs.
To avoid this problem, symmetric space is (equivalent to) a smooth manifold $M$ with multiplication $\cdot : M\times M\to M$ which is a smooth map such that for all $x,y,z\in M$
This amounts to an involutory quandle object $Q$ in the category of smooth manifolds, with the property that each point $a \in Q$ is an isolated fixed point of the map $a \triangleright - : Q \to Q$.
The definition in terms of quandles coincides with the classical definition in the case of connected symmetric spaces. For details, including a comparison of other definitions of symmetric space, see:
The relation to quandles is given in Theorem I.4.3. Bertram attributes this result to part I, chapter II of