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:

1. Fix a category $\mathcal{C}$.
2. 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.
3. 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.
4. To each object $U$ of $\mathcal{C}$ and a cover $\{U_\alpha\rightarrow U\}$, 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 $U$.
5. 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:

1. The category $\text{Sch}/S$ of schemes over a scheme $S$.
2. The category of manifolds $\text{Man}$.
3. 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?

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:

1. Fix a category $\mathcal{C}$.
2. 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.
3. 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.
4. 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 $U$.
5. 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:

1. The category $\text{Sch}/S$ of schemes over a scheme $S$.
2. The category of manifolds $\text{Man}$.
3. 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?

Are there any (What are the) occurrences of the notion of "stack" outside algebraic geometry?

In most of the references, the introduction of the notion of a stack takes the following steps:

1. Fix a category $\mathcal{C}$.
2. 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.
3. 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.
4. To each object $U$ of $\mathcal{C}$ and a cover $\{U_\alpha\rightarrow U\}$, 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 $U$.
5. 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 restrict ourselves to one of the following categories, with an appropriate Grothendieck topology:

1. Fix a scheme $S$ and consider the category $\text{Sch}/S$.
2. Category of manifolds $\text{Man}$.
3. 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:

Are there any (What are the) occurrences of the notion of "stack" outside algebraic geometry (other than what I have mentioned above)?

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:

1. Fix a category $\mathcal{C}$.
2. 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.
3. 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.
4. To each object $U$ of $\mathcal{C}$ and a cover $\{U_\alpha\rightarrow U\}$, 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 $U$.
5. 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:

1. The category $\text{Sch}/S$ of schemes over a scheme $S$.
2. The category of manifolds $\text{Man}$.
3. 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?

Are there any (What are the) occurrence of the notion of "stack" outside algebraic geometry?

In most of the references, introduction of the notion of a stack takes the following steps:

1. Fix a category $\mathcal{C}$.
2. 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.
3. Fix a Grothendieck topology on $\mathcal{C}$; this associates for each object $U$ of $\mathcal{C}$, a collection $\mathcal{J}_U$ (that is a collection of collection of arrows whose target is $U$) that are required to satisfy certain conditions.
4. For each object $U$ of $\mathcal{C}$ and a cover $\{U_\alpha\rightarrow U\}$; 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 $U$.
5. 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 restrict our self to one of the following categories:

1. Fix a scheme $S$ and consider the category $\text{Sch}/S$, and an appropriate Grothendieck topology on $\text{Sch}/S$.
2. Category of manifold $\text{Man}$ and an appropriate Grothendieck topology on $\text{Man}$.
3. Category of topological spaces $\text{Top}$, and an appropriate Grothendieck topology on $\text{Top}$.

Frequency of occurrence of stacks over above categories is in the decreasing order of magnitude. Unfortunately, I myself has seen exactly four research articles talking about stacks over the category of topological spaces.

So, the following question arises:

Are there any (What are the) occurrences of the notion of "stack" outside algebraic geometry (other than what I have mentioned above)?

Are there any (What are the) occurrences of the notion of "stack" outside algebraic geometry?

In most of the references, the introduction of the notion of a stack takes the following steps:

1. Fix a category $\mathcal{C}$.
2. 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.
3. 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.
4. To each object $U$ of $\mathcal{C}$ and a cover $\{U_\alpha\rightarrow U\}$, 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 $U$.
5. 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 restrict ourselves to one of the following categories, with an appropriate Grothendieck topology:

1. Fix a scheme $S$ and consider the category $\text{Sch}/S$.
2. Category of manifolds $\text{Man}$.
3. 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:

Are there any (What are the) occurrences of the notion of "stack" outside algebraic geometry (other than what I have mentioned above)?