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)
This is a sub-entry for gerbe.
For related entries see
Gerbes give a nice way to group together bundle data on a smooth manifold, but gerbes also naturally define degree two cohomology. Thus the idea of using gerbes in differential geometry is to have a nice language that relates geometric concepts such as connections and curvature to cohomological classifications.
In addition, one can use the analogies above that are made precise with gerbes to define other new concepts such as 3-curvature and “local” fiberwise connections.
Let $X$ be a smooth manifold. A Dixmier-Douady sheaf of groupoids over $X$ is a $\underline{\mathbb{C}}_X^*$- gerbe on $X$ where $\underline{\mathbb{C}}_X^*$ is the sheaf of smooth $\mathbb{C}^*$-valued functions (not to be confused with the constant sheaf $\mathbb{C}_X^*$).
We define $\mathbb{Z}(1)$ to be the term in the exponential sequence on $X$: $0\to \mathbb{Z}(1)\to \underline{\mathbb{C}}_X \to \underline{\mathbb{C}}_X^*\to 0$.
Taking the associated sequence in cohomology to the exponential sequence gives us an isomorphism $H^2(X, \underline{\mathbb{C}}_X^*)\overset\sim\to H^3(X, \mathbb{Z}(1))$.
We have a canonical isomorphism between the group of equivalence classes of Dixmier-Douady sheaves of groupoids over $X$ (basically by the definition of $\mathcal{A}$-gerbe) and $H^2(X, \underline{\mathbb{C}}_X^*)\simeq H^3(X, \mathbb{Z}(1))$.
Idea: Classes in $H^3(X, \mathbb{Z}(1))$ correspond to principal $G$- bundles over $X$ where $G$ is the projective linear group of a separable Hilbert space, namely $C^\infty (\mathbb{T})$.
Matt: Actually, a slight issue has arisen. Most of the things I thought would go here actually already appear in other places even though they aren’t grouped as coming from the same idea.
For instance, bundle gerbe contains the geometric interpretation of $H^3(X, \mathbb{Z}(1))$. Also, 3-curvature and fiber-wise connections occur at connection on a bundle gerbe. Although I think there is still a lot to say, I’m not convinced that “gerbe (in differential geometry)” is necessary anymore…
Jean-Luc Brylinski Loop Spaces, Characteristic Classes, and Geometric Quantization
I. Moerdijk, Introduction to the language of stacks and gerbes (arXiv)
Larry Breen, Notes on 1- and 2-gerbes (arXiv)
Further references are given in the other entries on gerbes.