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)
∞-Lie theory (higher geometry)
For $X$ a smooth space, there are useful refinements of the fundamental groupoid $\Pi_1(X)$ which remember more than just the homotopy class of paths, i.e. whose morphisms are (piecewise, say) smooth paths in $X$ modulo an equivalence relation still strong enough to induce a groupoid structure, but weaker than dividing out homotopies relative to endpoints.
Let $X$ be a smooth manifold.
For $\gamma_1, \gamma_2 : [0,1] \to X$ two smooth maps, a thin homotopy $\gamma_1 \Rightarrow \gamma_2$ is a smooth homotopy, i.e. a smooth map
with
which is thin in that it doesn’t sweep out any surface: every $2$-form pulled back to it vanishes:
A path $\gamma\colon [0,1] \to X$ has sitting instants if there is a neighbourhood of the boundary of $[0,1]$ such that $\gamma$ is locally constant restricted to that.
The path groupoid $P_1(X)$ is the diffeological groupoid that has
Composition of paths comes from concatenation and reparameterization of representatives. The quotient by thin-homotopy ensures that this yields an associative composition with inverses for each path.
This definition makes sense for $X$ any generalized smooth space, in particular for $X$ a sheaf on Diff.
Moreover, $P_1(X)$ is always itself naturally a groupoid internal to generalized smooth spaces: if $X$ is a Chen space or diffeological space then $P_1(X)$ is itself internal to that category. However, even if $X$ is a manifold, $P_1(X)$ will not be a manifold, see smooth structure of the path groupoid for details.
There are various generalizations of the path groupoid to n-groupoids and ∞-groupoids. See
If $G$ is a Lie group, then internal (i.e. smooth) functors from the path groupoid to the one-object Lie groupoid corresponding to $G$ are in bijection to $Lie(G)$-valued differential forms on $X$. With gauge transformations regarded as morphisms between Lie-algebra values differential forms, this extends naturally to an equivalence of categories
where on the left the functor category is the one of internal (smooth) functors.
More generally, smooth anafunctors from $P_1(X)$ to $\mathbf{B}G$ are canonically equivalent to smooth $G$-principal bundles on $X$ with connection:
See also