Does anyone know a reference for an explicit construction of 2-stackification (or even just (2,1)-stackification) by applying the plus construction three times, together with a complete proof that this actually works? I have found lots of places where this is asserted without proof (or with some sketch of a proof but many details omitted), and some places where the stackification is constructed differently (e.g. by first sheafifying the homs and then applying the plus-construction once), but nowhere that writes out the triple plus construction with all the details.

]]>Removed duplicate sentence.

]]>I have expanded the pointer to Giraud’s original text as follows:

Jean Giraud,

*Cohomologie non abélienne*, Columbia University (1966) (GoogleBooks)published as: Grundlehren 179, Springer Verlag (1971) (doi:10.1007/978-3-662-62103-5)

By the way, Artin’s stacks were introduced in

- Michael Artin,
*Versal deformations and algebraic stacks*, Inventiones Mathematics 27 (1974) 165-189

Urs, of course you are right that Deligne and Mumford were fully aware of Grothendieck-Giraud work on descent and stacks via fibered categories. Refering in Grothendieck’s school was mainly to give the most systematic reference rather than the origin of the first idea. For example, pro-objects were in print a bit before printed SGA but everybody refers to SGA4. Authors are anyway Grothendieck and Verdier. I refer for example to Duskin’s 1966/7 seminar published by Springer, and refering to J.-L.Verdier, Equivalence essentielle des systèmes projectifs, C.R.Acad.Sci 261 (1965), 4950-4953 and in a part of the text to Grothendieck. Again, the notion is from early 1960s.

]]>Giraud’s thesis under Grothendieck is written in 1966, and most of his later book is already there.

- J. Giraud, Cohomologie non abélienne de degré 2, thèse, Paris (1966).

FGA does not have stacks, but it does have effective descent and effective descent morphisms which along with definition of Grothendieck topologies, also from (another issue of) FGA are ingredients of the definition of a stack. According to VIstoli, (French) words champs (and prechamps (in the original sense, now used only in 1-categorical case, with sheaf conditions for homs) are both due Grothendieck. Giraud himself in his paper Méthode de la descente 1964 (doi) talks about the morphisms of 1-descent and 2-descent refering to the cases when the comparison functor along the morphism is either fully faithful or equivalence of categories (the latter now said to be of effective descent).

SGA1 was held n 1960/1961 and has both the theory of fibered categories and a little about stacks as well; the published version is as late as 1971 however and it refers to Giraud for more details. See the updated version of SGA 1 on arXiv:/math/0206203, page 259 (original page 345, file page 275). It is expose XIII of Mme Raynaud, according to the original (I guess 1961?) seminar notes of Grothendieck.

]]>Let’s see, Deligne-Mumford in their article do attribute the concept to Giraud’s book

On p. 24 they write:

We propose the terminology “stack” for the French word “champ” of non-abelian cohomology (Giraud [G])

However, their reference [G] doesn’t carry a date, it appears in full as

GIRAUD,

Cohomologie non abelienne, University of Columbia

So maybe it was available to them but not published yet at that time.

In any case, I’ll add their article to the entry on stacks, thanks.

Which page in FGA should I look at?

]]>Deligne-Mumford paper is from 1969, refining in algebro-geometric context on earlier Grothendieck’s ideas.

The descent condition and basic descent theorems used in the context of fibered categories/pseudofunctors are from Grothendieck’s FGA 1959 (numdam), and Grothendieck-Gabriel SGA I, 1961. GIraud was Grothendieck’s student and his original contribution were more specifically gerbes, I think.

]]>added early references:

The concept originates, under the French term *champ*, in

- Jean Giraud, _Cohomologie non abélienne, Grundlehren 179, Springer Verlag (1971) (doi:10.1007/978-3-662-62103-5)

and under the English term *stack* in

- Jean Giraud,
*Classifying topos*, in: William Lawvere (ed.)*Toposes, Algebraic Geometry and Logic*, Lecture Notes in Mathematics, vol 274. Springer (1972) (doi:10.1007/BFb0073964)

Further early discussion includes

Marta Bunge, Robert Pare,

*Stacks and equivalence of indexed categories*, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 20 no.4 (1979) (numdam)Marta Bunge,

*Stack completions and Morita equivalence for categories in a topos*, Cahiers de topologie et géométrie différentielle xx-4, (1979) 401-436, (MR558106, numdam)

added pointer to

- John F. Jardine,
*Stacks and the homotopy theory of simplicial sheaves*, Homology, homotopy and applications, vol. 3 (2), 2001, pp.361–384 (euclid:hha/1139840259)

This entry has loads of room to be (re-)written. Volunteers are welcome.

But (2,1)-sheaves are in particular 2=(2,2)-sheaves, and not the other way around, and that’s what the first two sentences refer to.

]]>"The term stack, is a traditional synonym for 2-sheaf or often just (2,1)-sheaf (see there for more details).

It is also often used more restrictively as a synonym for (2,1)-sheaf."

The first sentence seems to imply that 2-sheaf and (2,1)-sheaf are synonyms. The second sentence implies that they are not. In any case, the second sentence repeats something the first sentence was trying to say. Perhaps that could be clarified by combining the sentences. ]]>

That is of course not the famous Giraud’s book Cohomologie non abelienne but his earlier Memoirs SMS article on descent, which is as a size of a small book as well and which contributed in its content to the later book. I will teach descent theory and nonabelian cohomology in a graduate course the next academic year.

]]>Giraud is available on Numdam.

]]>Yeah, like the sheaf condition in terms of the category of descent data for a stack. I have a copy of Giraud’s original book where I believe they were introduced if you want a classic presentation.

]]>My aim was the first of those. The point is to put some flesh on the description of constant and locally constant (and eventually construcible ) stacks that is given by Treumann in his paper on exit paths. (My direction after that would be towards the Lurie version but I wanted the locally constant and constructible cases of 1-stacks done in a way that would be understandable to someone without enormous $\infty$-categorical knowledge.) One of Treumann’s discussions needed ’stalks’ and that if well done using pullbacks etc. Filling in details of the induced adjointness between Stacks(X) and Stacks(y) seemed one way to proceed. Any thought could be useful. The end result is to try to incorporate ‘defects’ into TQFTs and HQFTs with a bit more clarity (for me!) than at present exists in the literature. (This is a follow on to things talked on at the Lisbon meeting.)

]]>What sort of base change? Like having a stack on a space $X$ and then pulling back along $Y\to X$? Or do you mean the functors between fibres induced by maps in the site?

]]>The entry on stacks contains the request:

Somebody should turn this here into a coherent entry on stacks.

(This was already in Revision 4, from 2009!)

I was looking down a down to earth discussion and noted the old sketch that Todd gave. This is useful but does someone have the time / inclination to edit to get something more in line with the other entries. I really wanted something looking at base change for stacks that could be used in introductory notes, i.e. more approachable to non-categorists. Any thoughts?

]]>I added an MO link to the connection between groupoids and stacks. Is this the best reference?

]]>Okay, maybe you could add that remark.

]]>Stack entry says: "The notion of stack is the one-step vertical categorification of a sheaf." In Grothendieck's main works, like pursuing stacks and in the following works of French schools, stack is any-times categorification of a sheaf, and the one-step case is called more specifically 1-stack. We can talk thus about stack in narrow sense or 1-stacks and stacks in wider sense as n-stacks for all n. Topos literature mainly means that the stack is the same as internal 1-stack.

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