This entry sketches a very general abstract nonsense setup that is supposed to model the necessary and sufficient general structural context which admits models of fundamental (quantum) physics such as quantum field theory (gauge theory, sigma-models…) and string theory.
General as the setup is, we organize it into the primordial concepts space and process that accuratly reflect our two technical ingredients geometric structure and differential geometric structure and from that cohomology and differential cohomology.
The reader who finds this nonsense too general to be helpful should feel free to ignore it and follow the links to more concrete nonsense instead.
The spaces of relevance in many applications physics carry more structure than plain topological space, they carry certain geometric structure, specified by local model spaces.
This may be formalized by fixing a an (∞,1)-category $\mathcal{T}$ whose objects we think of as loci – test spaces with which all spaces with $\mathcal{T}$-geometry structure may be probed – and whose morphisms we think of maps between loci that respect the geometric structure in question.
Since the specification of $\mathcal{T}$ encodes what we want to mean by geometric structure, $\mathcal{T}$ is called a geometry (or rather a pregeometry).
By the general abstract nonsense of space and quantity, the most general notion of space modeled on the test objects in $\mathcal{T}$ is an ∞-stack on (pro-objects in) $\mathcal{G}$. We write
for a choice of (∞,1)-category of (∞,1)-sheaves on pro-objects in $\mathcal{T}$: the gros (∞,1)-topos of $\mathcal{T}$-geometric spaces. The choice of $\mathbf{H}$ on top of the choice of $\mathcal{T}$ encodes the notion locality of spaces modeled on $\mathcal{T}$.
The Yoneda embedding $\mathcal{T} \hookrightarrow Sh_\infty(\mathcal{T})$ ensures that every test space in $\mathcal{T}$ may canonically be regarded as a general space modeled on $\mathcal{T}$. When studying geometry it is of interest to refine this inclusion of very simple into very general spaces through a hierarchy of types of spaces of decreasing rigid geometric structure, for instance:
where
$Str(\mathcal{T})$ are the $\mathcal{T}$-structured (∞,1)-toposes: those $\mathcal{T}$-probeable spaces that have something like an underlying topological space in the generalized form of an underlying petit (∞,1)-topos which is equipped with structure sheaves of function quantities with values in objects of $\mathcal{T}$;
$Sch(\mathcal{T})$ are the $\mathcal{T}$-generalized schemes: those $\mathcal{T}$-structured spaces that are not only probeable by object of $\mathcal{T}$, but are locally isomorphic to objects in $\mathcal{T}$.
A model for fundamental physics typically involves
one nice space $X$ – for instance a $\mathcal{T}$-generalized scheme: the target space that models physical spacetime;
a collection of auxiliary spaces that are more general object of $\mathbf{H}$, such that
mapping spaces $Maps(\Sigma,X)$ of maps into $X$ – see process – differential structure, dynamics- differential structure, dynamics)
coefficient objects $A$ for cocycles $X \to A$ on $X$ that encodes background gauge fields on $X$ – see geometric cohomologyohomology).
Every (∞,1)-topos comes with its notion of nonabeliab cohomology.
For $\mathcal{T} = {*}$ the trivial geometry, this is the ordinary cohomology of Top. If instead $\mathcal{T}$ is some kind of smooth geometry, the corresponding cohomology of $\mathbf{H} = Sh_{(\infty,1)}(Pro(\mathcal{T}))$ is a flavor of smooth cohomology: it classifies not just topological principal ∞-bundles, but smooth $\infty$-bundles.
These $\infty$-bundles on $X$ encode the kinematics for physical objects propagating in $X$.
…
e.g. Spin structure, String structure, Fivebrane structure.
Fundamental (quantum) physics describes processes in spaces in the form of $d$-dimensional particles tracing out trajectories in a space.
Since all spaces are locally modeled on the test objects for the (pre)geometry $\mathcal{T}$, admissable geometric trajectories should be determined by the collection of geometric trajectories in each object of $\mathcal{T}$. Moreover, the boundary of a $k$-dimensional trajectory should be a $(k-1)$-dimensional trajectory and two $k$-dimensional trajectories should be composable along a joint boundary to a new $k$-dimensional trajectory. Finally, the collection of all trajectories should itself be a space modeled on $\mathcal{T}$.
This suggests that that a specification of geometric $d$-dimensional trajectories is encoded by a map
such that for $X \in \mathcal{T}$ the $(\infty,n+1)$-sheaf $\mathrm{Bord}_n(X)$ assigns to a test space $U \in \mathcal{G}$ an (∞,n)-category $Bord_n(X)(U)$ whose $k$-morphisms are $U$-families of $k$-dimensional trajectories in $X$. In particular to the point it assigns a version of the ordinary unstructured (∞,n)-category of cobordisms
The nature of fundamental (quantum) physics suggests that $Bord_n(X)$ should be such that for $k \lt n$ the composite of a $k$-dimensional trajectory with its reversed version is connected by a $(k+1)$-dimensional trajectory to the constant $k$-trajectory. This means in particular that we expect $Bord_\infty(X)$ to be a (stable symmetric) ∞-groupoid in that it is not just in $Sh_{(\infty,\infty)}(\mathcal{T})$ but actually in $Sh_{(\infty,1)}(\mathcal{T})$.
Finally, locality of quantum physics should imply in particular that all $k$-dimensional trajectories without boundary are obtained from gluing $k$-dimensional trajectories with boundary. This should mean that there is a smallest subcollection
of elementary trajectories such that all others are generated from these under gluing along common boundaries.
In summary we find that encoding a notion of processes in a space amounts to choosing the structure $\Pi$ of a $Pro(\mathcal{T})$-structured (∞,1)-topos on the gros $(\infty,1)$-topos $\mathbf{H} = Sh_{(\infty,1)}{Pro(\mathcal{T})}$ itself
The path ∞-groupoid is like a structure sheaf on a gros (∞,1)-topos.
This leads to a particularly symmetric situation of a structured (∞,1)-topos, where in fact we are dealing with bi-sheaves
In this special situation we have the Yoneda extension
of the path ∞-groupoid construction $\Pi(-)$ to a morphism
that computes the $\infty$-path in general $\mathcal{T}$-spaces. This has a right adjoint
The choice of such a geometric structure $\Pi : Pro(\mathcal{T}) \to \mathbf{H} = Sh_{(\infty,1)}(Pro(\mathcal{T}))$ on a gros $(\infty,1)$-topos we call a differential geometric structure or just differential structure.
On a gros $(\infty,1)$-topos $\mathbf{H}$ structured by a choice of path ∞-groupoid assignments $\Pi(-)$ we geometric cohomologyohomology) of $\mathbf{H}$ refines to a notion of
Differential cocycles on target space $X$ are what encodes gauge fields on $X$.
These $\infty$-bundles with connection on $X$ encode the dynamics for objects propagating in $X$.
… e.g. gauge fields, Background fields in twisted differential nonabelian cohomology, twisted differential String- and Fivebrane structures