TITLE: What are the occurrences of stacks outside algebraic geometry, differential geometry, and general topology?
QUESTION [20 upvotes]: 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?

REPLY [8 votes]: Mike Shulman has stack semantics, an application of stacks to logic.  This is basically sheaf semantics, a now standard application to logic of sheaves (far from their own origin in geometry), except that sheaf semantics isn't quite powerful enough to capture unbounded quantification in the way that Mike needs in order to do what he wanted to do with set theory (which is what he was doing when he came up with stack semantics).
This is a fairly low-powered application of stacks, as sheaves are almost but not quite sufficient.  But simply adopting this approach makes some things easier to talk about, even when one could do them in the old (sheaves-only) way.  And if you want to apply this kind of logic to category theory itself instead of to set theory, then the stacks are really necessary.