# What are the occurrences of stacks outside algebraic geometry, differential geometry, and general topology?

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?

• There would be of course more articles on topological stacks than what I have mentioned above.. Please let me know if you come across any other.. It is just that I have come across only four.. But, I am sure the number would be much less when compared to articles dealing with algebraic stacks.. – Praphulla Koushik May 30 '20 at 17:19
• I did some proofreading, hopefully without changing any meanings. Is it really true that $\mathcal J_U$ is a collection of collections of arrows, or is one of those layers a typo? – LSpice May 30 '20 at 17:24
• @LSpice It is not a typo.. It associates to each object a collection of collection of arrows... :D each collection in that collection is called a cover of $U$.. Thanks for other corrections.. – Praphulla Koushik May 30 '20 at 17:27
• The entire chapter of the book by Bridson and Haefliger is about nonpositively curved complexes of groups which are stacks in the category of simplicial complexes. Orbifolds are widely used in low dimensional topology. – Moishe Kohan May 30 '20 at 20:48
• Stack semantics in categorical semantics, no? – xuq01 May 31 '20 at 18:27

## 7 Answers

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.

• I might be able to understand this once I have some understanding of $\infty$-category (I started reading something about infinity categories, so it should not take much time)... Does any other examples comes to your thought that does not have anything to do with infty stacks and only uses just stacks (1-stacks)... – Praphulla Koushik May 31 '20 at 1:56
• @PraphullaKoushik: You can certainly consider stacks in groupoids over C^∞-rings. These do not have a good theory of all homotopy colimits, but they do have a good theory of ordinary quotients. – Dmitri Pavlov May 31 '20 at 2:33
• Ok. I am hearing about $C^{\infty}$-rings first time... stacks in groupoids over the category of $C^{\infty}$-rings (equipped with some nice Grothendieck topology)... I see there are some references about $C^{\infty}$-rings at ncatlab.org/nlab/show/smooth+algebra I will see them.. – Praphulla Koushik May 31 '20 at 3:35

Stacks are used in complex analysis, for example.

See the papers by Finnur Lárusson, in particular, Excision for simplicial sheaves on the Stein site and Gromov's Oka principle, which shows that having the Oka–Grauert property for a complex manifold X is equivalent to the condition that the presheaf of spaces of holomorphic maps into X is an ∞-stack in the appropriate Grothendieck topology on the site of Stein manifolds.

A few years ago, Bernstein wrote a note with a new approach to representation theory of algebraic groups using the langage of stacks.

• This doesn't sound to be like an occurence "outside algebraic geometry". – Wojowu May 31 '20 at 20:09
• @Wojowu : the note is supposed to give another approach to representation theory of algebraic group over local field (using stacks to consider different forms of a group "at once"), which seems outside algebraic geometry. – Nicolas Hemelsoet May 31 '20 at 20:18
• Thanks. I will see that paper. :) – Praphulla Koushik Jun 1 '20 at 2:46

Stacks over the category locales are very interesting for topos theory:

A big success of topos theory is the fact that the $$(2,1)$$-categories of Grothendieck toposes and geometric morphisms between them embedded as a reflective full subcategory of the category of localic stacks, that is stack on the category of locales. It is in fact a full subcategory of the category of "Geometric localic stacks", that is those localic stacks comming from localic groupoids.

In my mind this is the results that best convey the idea that Grothendieck toposes are geometric objects. Of course, Grothendieck had the intuition that toposes were geometric object from the very beginning of the theory, but this results is to me really what turn this intuition into something formal.

Note: There are some size issues involved whose discussion will be postponed to the very end.

We will identifies the category of locales with a full subcategory of the category of toposes, by identifying each locales $$\mathcal{L}$$ with the sheaf topos Sh$$(\mathcal{L})$$.

The basic idea is fairly simply given $$\mathcal{T}$$ a topos and $$\mathcal{L}$$ a locale, you get a category of geometric morphisms Hom$$(\mathcal{L},\mathcal{T})$$, if you simply drop the non-invertible natural transformations, you get a groupoid Hom$$(\mathcal{L},\mathcal{T})$$ of geometric morphisms and natural transformations.

This attach to every topos a pre-stack on the category of locales. It can be shown that this pre-stack is a stack for the topology whose coverings are the open surjections between locales (and the coproduct).

This construct a functor from the $$(2,1)$$-category of toposes to the $$(2,1)$$-category of localic stacks, which is fully faithful and identifies the category of toposes with a reflective full subcategory of stacks. Stack in the image are called "etale-complete" stack (to be honest one generally talks about étale-complete localic groupoids, but this is a property of the associated stack).

The starting point of this story started with the famous representation theorem of Joyal and Tierney in "An Extension of the Galois Theory of Grothendieck", which can be understood as the construction of the left adjoint, and the proof that it is essentially surjective, though most of the key idea are already present.

The results as presented above appeared in the two paper of Moerdijk:

The classying topos of a continuous groupoids, I & II

As the title suggest the results is mostly stated in terms of groupoids rather than stack, but the theory is really about stacks, and if I remember correctly the connection to stack is explicit mentioned in the paper. I think Bunge's paper "An application of descent to a classification theorem for toposes" is also relevant to the story.

So what I've said above is only correct up to some important size consideration that need to be taken care of.

The category of locales, with the topology of open surjections, do not satisfies the smallness condition needed in order for stackification to be well defined.

Though the point of view we adopt here, is that up to passing to a larger Grothendieck universe stackification is always defined, the question is only wether or not preserve it preserves certain smallness conditions.

In this case stackification do not preserve smallness: there are examples of small pre-stack of locales (in the sense "small colimits of representable") whose stackification is not even "levelwise small", that is $$\mathcal{F}(\mathcal{L})$$ can fail to be an essentially small groupoids.

But this is actually a good things, because for many Grothendieck topos, the groupoids Hom$$(\mathcal{L},\mathcal{T})$$ are not essentially small.

Here the appropriate "category of stack" to consider for what I say above to be correct are the large stack that are small colimits (in the category of stack) of representable stacks. This is not a locally small category (but the category of Grothendieck topos isn't either). The fact that the stack attached to a topos is in this category is non trivial, but follows directly from the work of Joyal and Tierney mentioned above.

• Thank you :) This is mode detailed than what I can expect.. I will read it carefully and ask if I have any questions :) – Praphulla Koushik Jun 2 '20 at 5:14

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.

There are two notions of stack. The one you mention is a sheaf of groupoids. Sometimes these come up on their own. The other notion is a geometric object, often a "bad quotient." This object can be represented as a sheaf of groupoids, but that is only a technical tool. If you had other tools, you might use them instead. For example, if you had a foliation of a manifold, you might want to consider the "space of leaves." You could consider this as a stack on the site of topological spaces, but you could also represent it by the convolution algebra of the equivalence relation. Constructions that are Morita invariant depend only on the stack. So you might say that Connes-style noncommutative geometry is (in part) the study of stacks, or you might say it is a reason that stacks are not more popular.

• I do not understand your last sentence "So you might say that Connes-style noncommutative geometry is (in part) the study of stacks, or you might say it is a reason that stacks are not more popular." Can you explain what it means? – Praphulla Koushik May 31 '20 at 17:53
• Groupoid convolution algebras can indeed be used to study sufficiently nice stacks (e.g., those presented by Lie groupoids), but claiming that convolution algebras are stacks is stretching things too far. For instance, is there an actual theorem (with proof) in the literature that establishes an equivalence of bicategories between sufficiently nice stacks and sufficiently nice algebras? Otherwise one simply cannot refer to algebras as a “notion of stack”. – Dmitri Pavlov May 31 '20 at 18:49
• @DmitriPavlov Yes, but it's an exercise. – Ben Wieland Jun 1 '20 at 0:36
• @BenWieland: If it is an exercise, can you tell us what algebras (with a precise list of properties) correspond to Lie groupoids and what bimodules between these algebras (again with a precise list of properties) correspond to bibundles between Lie groupoids? – Dmitri Pavlov Jun 1 '20 at 0:52
• @BenWieland: I know quite a few people who would be extremely interested in seeing a reference for the construction you are alluding to. This would solve a known research problem. For starters, what algebras correspond to Lie groupoids? And how do we characterize bimodules corresponding to bibundles? – Dmitri Pavlov Jun 2 '20 at 16:10

There are "bundle gerbes" (introduced by Murray), which are a particular kind of stacks. People study connections on them, generalizing connections on principal bundles.

• How is this an answer to OP's question? Bundle gerbes are stacks on the site Man of smooth manifolds, which is Example 2 in OP's list. The OP specifically requested an example not in his list. – Dmitri Pavlov May 31 '20 at 18:59
• @Dmitri Pavlov: by being an answer to the question in the title, and to the first question in the body of the OP. – Qfwfq May 31 '20 at 19:09
• @Dmitri Pavlov: but, upon a second reading, I realize the only question was likely intended to be the last one; the first one (along with the title) was probably to be intended as general context to be specified in the body of the OP. – Qfwfq May 31 '20 at 19:12
• Surely you can see that the two questions are identical, except for the clarifying parenthetical remark? Gerbes are already discussed in the papers cited by the OP in the main post (e.g., in Carchedi's paper), repeating examples from the main post as separate answers is a rather strange thing to do. – Dmitri Pavlov May 31 '20 at 19:26
• Yup: the parenthetical remark in the second question rules out the examples mentioned in the previous paragraphs; which the first question didn't. -- If you, or anyone, thinks my reading of the OP was off-topic (which may be the case), I'm totally fine with my answer being deleted. I'm not going to comment in this thread anymore. – Qfwfq May 31 '20 at 19:39