What are the occurrences of the notion of a stack outside algebraic geometry, differential geometry, and general topology?
In most of the references, the introduction of the notion of a stack takes the following steps:
- Fix a category $\mathcal{C}$.
- Define the notion of category fibered in groupoids/ fibered category over $\mathcal{C}$; which is simply a functor $\mathcal{D}\rightarrow \mathcal{C}$ satisfying certain conditions.
- Fix a Grothendieck topology on $\mathcal{C}$; this associates to each object $U$ of $\mathcal{C}$, a collection $\mathcal{J}_U$ (that is a collection of collections of arrows whose target is $U$) that are required to satisfy certain conditions.
- To each object $U$ of $\mathcal{C}$ and a cover $\{U_\alpha\rightarrow U\}$, after fixing a cleavage on the fibered category $(\mathcal{D}, \pi, \mathcal{C})$, one associates what is called a descent category of $U$ with respect to the cover $\{U_\alpha\rightarrow U\}$, usually denoted by $\mathcal{D}(\{U_\alpha\rightarrow U\})$. It is then observed that there is an obvious way to produce a functor $\mathcal{D}(U)\rightarrow \mathcal{D}(\{U_\alpha\rightarrow U\})$, where $\mathcal{D}(U)$ is the "fiber category" of $U$.
- A category fibered in groupoids $\mathcal{D}\rightarrow \mathcal{C}$ is then called a $\mathcal{J}$-stack (or simply a stack), if, for each object $U$ of $\mathcal{C}$ and for each cover $\{U_\alpha\rightarrow U\}$, the functor $\mathcal{D}(U)\rightarrow \mathcal{D}(\{U_\alpha\rightarrow U\})$ is an equivalence of categories.
None of the above 5 steps has anything to do with the set up of algebraic geometry. But, immediately after defining the notion of a stack, we typically restrict ourselves to one of the following categories, with an appropriate Grothendieck topology:
- The category $\text{Sch}/S$ of schemes over a scheme $S$.
- The category of manifolds $\text{Man}$.
- The category of topological spaces $\text{Top}$.
Frequency of occurrence of stacks over above categories is in the decreasing order of magnitude. Unfortunately, I myself have seen exactly four research articles (Noohi – Foundations of topological stacks I; Carchedi – Categorical properties of topological and differentiable stacks; Noohi – Homotopy types of topological stacks; Metzler – Topological and smooth stacks) talking about stacks over the category of topological spaces.
So, the following question arises:
What are the occurrences of the notion of a stack outside of the three areas listed above?
Best Answer
Another application of stacks is in synthetic differential geometry.
Start with the opposite category of germ-determined finitely generated C^∞-rings and equip it with the appropriately defined Grothendieck topology, then pass to ∞-stacks.
The resulting category (known as the Dubuc topos) contains all smooth manifolds, is a Grothendieck ∞-topos (so in particular, has all homotopy colimits and is cartesian closed), and allows for a good notion of infinitesimals. The latter allows to manipulate differential geometric objects such as vector fields and differential forms using infinitesimal methods similar to the ones used by Élie Cartan and Sophus Lie, yet perfectly rigorous. For instance, the de Rham complex is now precisely the smooth infinitesimal singular cochain complex, and the Stokes theorem is now precisely the definition of the de Rham differential as the singular cochain differential. Just like for stacks on manifolds, homotopy colimits in this category have excellent geometric properties.
Even better, if one takes germ-determined finitely generated differential graded C^∞-rings and takes ∞-stacks on the resulting ∞-site, then one gets the ∞-stack that has all the excellent properties listed above, together with excellent geometric properties of homotopy limits (which always exist). In particular, in this category nontransversal intersections exist and have desired geometric properties, etc. This subject is known as derived differential geometry.