User jonathan chiche - MathOverflow

Answer by Jonathan Chiche for Conjectures in Grothendieck's "Pursuing stacks"

2012-12-06T09:36:57Z

This is more a comment than an answer, but its length makes me post it as an answer. I want to react to what I have just read, for the first time, about "Pursuing Stacks" at the nLab, and the words used there as well as in your question. I find it extremely irritating when people only use words such as "ideas" and "conjectures" to describe the content of "Pursuing Stacks". It makes me wonder whether they have read it or not. The words "rigorous", "results" or "theorems" are not used unless they describe other people's work. This is perfectly true that you would find ideas and conjectures in "Pursuing Stacks", but it does not prevent many notions and results regarding them to be "rigorously worked". It absolutely blows my mind that, nearly thirty years after the writing of this text, people still talk about it in that disrespectful way. (Those who do not think this is disrespectful to credit Grothendieck with "ideas" and "conjectures", no matter how "deep" or "beautiful", while others get the credit for the rigorous results, can read "Récoltes et semailles", where this question is precisely addressed at length.)

This is nevertheless true that "Pursuing Stacks" contain many ideas and conjectures, and David Roberts's answer gives, to my very partial knowledge, an accurate rough overview of what has been developed since then. I would just mention the fact that Grothendieck did not view simplicial sets as more homotopically relevant than other test categories. As regards $\infty$-groupoids, his approach is purely algebraic. Therefore, while I certainly do not claim that one of the approaches is better than the others, the prevalent simplicial approaches do not seem to me to be the one advocated by Grothendieck.

EDIT 1 (10 December 2012): This "answer" has got one downvote and someone has canceled their upvote. Perhaps I should have explained more. I find the question and the answers given so far quite sad because they all point out the fact that almost nobody has read "Pursuing Stacks", and people mostly talk about it by word of mouth. If some people had read it carefully, then sure enough they would have cited accurately some of Grothendieck's results. What are these precises references in the literature? I only know that there was a working group organized by Bénabou in the eighties, devoted to Pursuing Stacks, soon after its sending, but it quickly came to an end, and that someone in the Netherlands asked a student, whose name nobody seems to recall, to work on test categories, which he did not pursue very far. There certainly has been a lot of work in "categorical homotopy theory" since the writing of "Pursuing Stacks", but have Lurie, Rezk, Toën, Vezzosi (all cited by David Roberts) read "Pursuing Stacks" (I certainly do not claim they should have)? I think one should be very careful before claiming that one's own work or other people's work is indeed related to the content of "Pursuing Stacks" and even more before stating that it "realize" (?) or "formalize" some "ideas" of "Pursuing Stacks". To some extent, this text is like the Bible: eveybody talks about it but almost nobody knows what is written there.

What has happened to Lang's Files and other political texts?

2012-03-08T10:33:06Z

For some background on Lang and his files, one can read the first part of Lang's obituary in the AMS Notices at http://www.ams.org/notices/200605/fea-lang.pdf.

The book "Challenges" was published in 1997. I guess Lang went on to gather documentation regarding several cases dealt with in the book after its publication. Consequently, several related files must have contained much more documentation at the time of Lang's demise than at the time of "Challenges" publication. Besides, "Challenges" does not deal with all the cases Lang has been involved with and interested in.

In addition, the AMS obituary mentions that Lang "has some unpublished books of a political nature".

Are the full (i.e. the whole documentation that Lang had gathered and arranged) files regarding the "Challenges" cases (i.e. those dealt in that book) published somewhere? If not, are they available in some other form? What has happened to Lang's related personal papers?

What about other files not reproduced in "Challenges"? What are they, and where is the documentation?

Same question for Lang's "unpublished books of a political nature".

EDIT 1 (27 November 2012). For some bad reasons, it has taken me several months to decide to write to the Yale librarian. I can now provide a partial answer to my question, that is: these papers have not been deposited at Yale's library. I will follow the librarian's suggestion to write to the University of Texas. If someone there (or elsewhere) reads this post and wants to provide help to locate Lang's papers, please do not hesitate to contact me. I have copied below the message I have received from Yale's librarian following my query. I have hidden the name of the sender, although that may be unnecessary.

November 26, 2012

Dear Mr. Chiche:

I am writing in reply to your e-mail inquiry of October 22, 2012, regarding former Yale professor Serge Lang.

Our on-line catalog shows that the Yale University Library holds a number of Professor Lang’s published works but, if I am interpreting your query correctly, you wish to know about his unpublished writings. The Manuscripts and Archives department does not hold a collection of Professor Lang’s papers. They were offered to us following his death but we declined since the subject matter was outside our collecting areas. We suggested the University of Texas, where a number of mathematical collections are held. I searched OCLC’s World Cat to see if I could locate the papers but I did not find any results. You may wish to contact the library directly to see if the Serge Lang Papers were ever received. Information and assistance is available at http://www.cah.utexas.edu/collections/math.php

If you have any additional questions, please do not hesitate to contact us.

Sincerely,

XXX

Archivist

Manuscripts and Archives

What should be the terminology regarding symmetries in 2-Cat?

2012-11-10T17:50:36Z

This is a boring question about $2$-categorical terminology.

When I began to learn about $2$-categories, I read papers using the words "lax functors", "oplax functors", "lax transformation" and "oplax natural transformation".

The first time I got the feeling something was wrong with that terminology was when I realized that the meanings of "lax" in "lax functor" and in "lax transformation" do not seem to be the same. For instance, if $u$ and $v$ are lax functors, say from $\mathcal{A}$ to $\mathcal{B}$, any "lax transformation" from $u$ to $v$ gives rise to a lax functor $h : \Delta[1] \times \mathcal{A} \to \mathcal{B}$ making the "obvious" diagram commute, i.e. there is some kind of "elementary (lax) homotopy" from $u$ to $v$. But the same conclusion holds if we replace "lax transformation" with "oplax transformation"! Therefore, an "oplax transformation" gives rise to a "lax homotopy". That seems pretty awkward to me. Is there a reason why one should use this terminology? Why does one not merely say "transformation" and "optransformation"?

In fact, there is worse. According to the prevalent terminology, the classical "dual notion" to lax functors are "oplax functors". But most people seem happy with the convention according to which ${?}^{op}$ denotes $?$ after inversion of the $1$-cells and ${?}^{co}$ denotes $?$ after inversion of the $2$-cells. Since "oplax functors" are nothing else than lax functors from $\mathcal{B}^{co}$ to $\mathcal{A}^{co}$, why should they be called "oplax"? "Colax" seems quite a better word from this viewpoint. I have just noticed that the nLab makes this choice. Perhaps some people here could tell me the history of both words and what is the current view towards them, at least in the anglo-saxon world.

Note that "optransformation" (or "oplax transformation", for that matter) are consistent with this convention, since this is nothing but a ("lax") transformation from $v^{op}$ to $u^{op}$. However, in ordinary category theory (i.e. $1$-category theory), the duality (which is taken with respect to $1$-cells, since there is no $2$-cells) is denoted by the prefix "co-", as in "comonads", "colimits", &c. This is not consistent with the $2$-categorical convention that the process of reversing $1$-cells should be denoted by "op-".

Where is it rigorously stated and proved that the definition of lax functor implies that the generalized cocycle condition holds for an arbitrary number of composable $1$-cells?

2012-01-07T08:32:15Z

Let $\mathcal{A}$ and $\mathcal{B}$ be two $2$-categories and $F : \mathcal{A} \to \mathcal{B}$ be a lax $2$-functor. Given $1$-cells $(f_{i})_{0 \leq i \leq n}$ of $\mathcal{A}$ such that the composition $f_{n} \circ f_{n-1} \circ \cdots \circ f_{0}$ makes sense, this data together with the structural $2$-cells of $F$ give many paths of $2$-cells going from $F(f_{n}) \circ F(f_{n-1}) \circ \cdots \circ F(f_{0})$ to $F(f_{n} \circ f_{n-1} \circ \cdots \circ f_{0})$, for instance $$ F(f_{n}) \circ F(f_{n-1}) \circ \cdots \circ F(f_{0}) \Rightarrow F(f_{n} \circ f_{n-1}) \circ F(f_{n-2}) \circ \cdots \circ F(f_{0}) \Rightarrow \cdots $$ $$\Rightarrow F(f_{n} \circ f_{n-1} \circ \cdots \circ f_{0}) $$ and $$ F(f_{n}) \circ F(f_{n-1}) \circ \cdots \circ F(f_{0}) \Rightarrow F(f_{n}) \circ F(f_{n-1}) \circ \cdots \circ F(f_{1} \circ f_{0}) \Rightarrow \cdots $$ $$\Rightarrow F(f_{n} \circ f_{n-1} \circ \cdots \circ f_{0}) $$ which correspond to what one gets by "parenthesizing on the left" and "parenthesizing on the right" respectively. It seems to seem obvious that it follows from the definition of lax functor that the $C_{n}$ ways to parenthesize the left hand side all give the same $2$-cell $$ F(f_{n}) \circ F(f_{n-1}) \circ \cdots \circ F(f_{0}) \Rightarrow F(f_{n} \circ f_{n-1} \circ \cdots \circ f_{0}) $$ Since I need this property for a text I am writing, I would like to provide a reference. My question is the following:

Where is this result rigorously stated, and where is it rigorously proved? Hopefully, the two references will be the same.

Edit: I am aware that this result is "obvious". In addition, it is certainly classical, by which I mean that all the people working with lax functors use it routinely. However, if one wants to state it and prove it, the question arises as to what is the best way to state the result, which I think turns out not to be completely trivial. Furthermore, writing a rigorous proof certainly does require some work. I am sure there are some people here who have already used this result. How do they state it? To which reference do they point? Or is the reader assumed to find this fact so obvious that no one ever cares to provide a proof or a reference?

Answer by Jonathan Chiche for A (too easy) normalization of a lax-funtor between 2-categories ?

2012-11-01T10:16:22Z

It seems to me that your question woud be clearer if you stated more precisely the axioms for lax functors. Anyway, while I am unsure as to what you have in mind, I guess that "lax functors" and "normal lax functors" have the same meaning for you and me. You may therefore be interested in Lemma 4.2. and the beginning of the proof of Theorem 6.3. of the paper Nerves and classifying spaces for bicategories by Carrasco, Cegarra and Garzón. Hope this helps.

Is there a standard name for a 2-category which has an object z such that, for every object x, the category Hom(x,z) has a terminal object?

2012-09-29T09:10:35Z

Motivation

In Pursuing Stacks, Grothendieck defines what he calls a basic localizer, which is, to put it roughly, a class of functors between small categories with which one can make homotopy in $Cat$. One axiom of basic localizers asserts that every category which has a terminal object is "aspherical", i.e. the canonical arrow from it to the point is in any basic localizer. It is a sufficient condition to get all the properties we want, provided the other axioms hold.

I have recently worked on "2-basic localizers", the analogous classes of (let say strict for convenience) 2-functors. It appears that what seems, from this viewpoint, the right analogous notion of "category with terminal object" is "2-category which has an object $z$ such that, given any object $x$, the category $Hom (x,z)$ has a terminal object". Instances of such 2-categories are "slices over an object". (There are, of course, three dual notions, the four of them corresponding to the two ways to slice over an object and the two ways to slice under an object.)

Question 1

Does this property have a standard name?

I am aware that there should be something like an adjunction between such a 2-category and the terminal one going on here, but I am really looking for standard terminology if there is one now.

Question 2

Do this kind of 2-categories or this kind of property appear naturally in other contexts?

This second question may be as silly as asking "where do categories with a terminal object crop up", but category theorists I have talked to do not seem to have encountered such a notion. I hope it could ring a bell for other people, especially those working on homotopy-related stuff.

EDIT: To be a bit more precise, I have the feeling that this property has something to do with prefibrations in $2-Cat$. Perhaps I will tell more about that later.

What is the history of the notion of subdivision of categories?

2012-07-27T08:01:52Z

A recent answer by Peter May prompts me to ask a question which I have been considering to ask for several months. (The reason why I have not asked it before is that it is not directly related to my research and that I therefore almost do not have any motivation but sheer curiosity.)

According to Peter May, there was a "folklore" notion of categorical subdivision in the 1960's. I learnt about it by Matias del Hoyo's paper cited by Roman Bruckner in his comment to Peter May's answer. If I am mot mistaken, this notion had appeared in Anderson's paper "Fibrations and Geometric Realizations" as well as a paper authored by Dwyer and Kan, "Function complexes for diagrams of simplicial sets".

Who introduced the notion of subdivision of a (small) category? Are there any early references other than the two aforementioned papers?

Del Hoyo claims that performing the subdivision of a small category amounts to taking the nerve, applying Kan's simplicial subdivision functor, and coming back in $Cat$ by applying nerve's left-adjoint. Unfortunately, he does not prove this result. (I have discussed about this fact with him recently. If my memory serves me right, among other things, he proves that Anderson's and Dwyer-Kan's notions are equivalent.) Georges Maltsiniotis has given a rough proof of the verification to me. I was at that time unable to find any published proof. Even if it is an easy "folklore" result, I think it would be useful to have a proof publicly available somewhere.

Is there a published proof of the fact that this categorical subdivision is merely the composition of three well-known functors as above? Was it also "common knowledge" in the 1960's?

Finally, I cannot help asking a question which had come to my mind at that time, but to which I have not devoted much consideration since then. Using higher categorical nerves, there is an "obvious" definition of what could be analogs of this construction for higher categories. Therefore:

Have "higher analogs" of this categorical subdivision been studied?

Answer by Jonathan Chiche for Is there a topos theoretic interpretation/proof of Quillen's Theorem A?

2012-06-04T09:15:51Z

I have no compelling answer to this question myself, but you may find relevant results and ideas in the work of Grothendieck, Maltsiniotis and Cisinski in homotopical algebra. Have you looked at Pursuing Stacks? In Maltsiniotis's Astérisque, there is a hint as to what a cohomological proof of Quillen's result would be. See http://www.math.jussieu.fr/~maltsin/ps/prstnew.pdf, page numbered 11 in the document. The text of Cisinski's Astérisque is available at http://www.math.univ-toulouse.fr/~dcisinsk/publications.html. It is called Les préfaisceaux comme modèles des types d'homotopie. He also has an old and never-published preprint —which may contain some typos— called Faisceaux localement asphériques, available at the same webpage. I hope this helps and is not off the point.

In which situations can one see that topological spaces are ill-behaved from the homotopical viewpoint?

2011-09-27T13:11:07Z

In the eighties, Grothendieck devoted a great amount of time to work on the foundations of homotopical algebra.

He wrote in "Esquisse d'un programme": "[D]epuis près d'un an, la plus grande partie de mon énergie a été consacrée à un travail de réflexion sur les fondements de l'algèbre (co)homologique non commutative, ou ce qui revient au même, finalement, de l'algèbre homotopique." (Beginning of section 7. English version here: "Since the month of March last year, so nearly a year ago, the greater part of my energy has been devoted to a work of reflection on the foundations of non-commutative (co)homological algebra, or what is the same, after all, of homotopic[al] algebra.)

In a letter to Thomason written in 1991, he states: "[P]our moi le “paradis originel” pour l’algèbre topologique n’est nullement la sempiternelle catégorie ∆∧ semi-simpliciale, si utile soit-elle, et encore moins celle des espaces topologiques (qui l’une et l’autre s’envoient dans la 2-catégorie des topos, qui en est comme une enveloppe commune), mais bien la catégorie Cat des petites caégories, vue avec un œil de géomètre par l’ensemble d’intuitions, étonnamment riche, provenant des topos." [EDIT 1: Terrible attempt of translation, otherwise some people might miss the reason why I have asked this question: "To me, the "original paradise" for topological algebra is by no means the never-ending semi-simplicial category ∆∧ [he means the simplex category], for all its usefulness, and even less is it the category of topological spaces (both of them imbedded in the 2-category of toposes, which is a kind of common enveloppe for them). It is the category of small categories Cat indeed, seen through the eyes of a geometer with the set of intuitions, surprisingly rich, arising from toposes."]

If $Hot$ stands for the classical homotopy category, then we can see $Hot$ as the localization of $Cat$ with respect to functors of which the topological realization of the nerve is a homotopy equivalence (or equivalently a topological weak equivalence). This definition of $Hot$ still makes use of topological spaces. However, topological spaces are in fact not necessary to define $Hot$. Grothendieck defines a basic localizer as a $W \subseteq Fl(Cat)$ satisfying the following properties: $W$ is weakly saturated; if a small category $A$ has a terminal object, then $A \to e$ is in $W$ (where $e$ stands for the trivial category); and the relative version of Quillen Theorem A holds. This notion is clearly stable by intersection, and Grothendieck conjectured that classical weak equivalences of $Cat$ form the smallest basic localizer. This was proved by Cisinski in his thesis, so that we end up with a categorical definition of the homotopy category $Hot$ without having mentionned topological spaces. (Neither have we made use of simplicial sets.)

I personnally found what Grothendieck wrote on the subject quite convincing, but of course it is a rather radical change of viewpoint regarding the foundations of homotopical algebra.

A related fact is that Grothendieck writes in "Esquisse d'un programme" that "la "topologie générale" a été développée (dans les années trente et quarante) par des analystes et pour les besoins de l'analyse, non pour les besoins de la topologie proprement dite, c'est à dire l'étude des propriétés topologiques de formes géométriques diverses". ("[G]eneral topology” was developed (during the thirties and forties) by analysts and in order to meet the needs of analysis, not for topology per se, i.e. the study of the topological properties of the various geometrical shapes." See the link above.) This sentence has already been alluded to on MO, for instance in Allen Knutson's answer there or Kevin Lin's comment there.

So much for the personal background of this question.

It is not new that $Top$, the category of all topological spaces and continuous functions, does not possess all the desirable properties from the geometric and homotopical viewpoint. For instance, there are many situations in which it is necessary to restrict oneself to some subcategory of $Top$. I expect there are many more instances of "failures" of $Top$ from the homotopical viewpoint than the few I know of, and I would like to have a list of such "failures", from elementary ones to deeper or less-known ones. I do not give any example myself on purpose, but I hope the question as stated below is clear enough. Here it is, then:

In which situations is it noticeable that $Top$ (the category of general topological spaces and continuous maps) is not adapted to geometric or homotopical needs? Which facts "should be true" but are not? And what do people usually do when encountering such situations?

As usual, please post only one answer per post so as to allow people to upvote or downvote single answers.

P.S. I would like to make sure that nobody interpret this question as "why should we get rid of topological spaces". This, of course, is not what I have in mind!

Is there any elementary text unravelling the definitions of 2-category, lax functor and lax transformation, allowing people who do not know in the first place what these things are to really understand the definitions?

2011-11-25T11:44:49Z

The question is in the title.

My current research subject is the homotopy theory of $2$-categories. It may be slightly unreasonable to expect my PhD thesis to be read or even looked at by people outside the field of higher categories, but for some reasons I would like this text to be enough self-contained so as to allow someone without any prior knowledge of $2$-category theory to understand at least the basic definitions. "Someone without any prior knowledge of $2$-category theory" does not refer to someone not knowing anything about category theory. However, I do not want to assume the reader to know anything more advanced than the very basic definitions of category theory. I therefore would like to be able to point out the reader to a text which does not require more advances prerequisites than elementary category theory. Otherwise I shall write the full definitions myself if no such text is available. There are some texts introducing $2$-categories in the literature, but all the references I have come across are ruled out because of the following requirements:

It really should contain the unravelled definitions of $2$-category, lax functor and lax transformation;

I do not want the word "enriched" to be used;

If the word "natural" is used, then the meaning of the naturality should be explained in full;

The vocabulary has to be consistent with current usage;

The $2$-categories have to be strict because I really do not make any use of any other objects. This condition is, however, less important than the previous ones, but it would be clumsy to write things like "for the unravelled definitions, see Leinster's "Basic Bicategories" and strictify the coherence conditions".

I think the "big.list" tag is not approppriate because I am really looking for a single reference which would fulfill these conditions.

EDIT: Thanks to those people who have commented on this question. Unfortunately, I think that none of the references provided gives what I was looking for. The closest I have found is Tom Leinster's "Basic Bicategories" which I had cited in the question, the rub being that it is an introduction to general bicategories, which implies coherence diagrams much more complicated than in the context of $2$-categories. The way to simplify them in this special case soon becomes obvious once one has acquired some experience, but I do not want to impose this task on a reader whom I shall assume without any prior acquaintance with $2$-categorical notions, let alone bicategories. The logical conclusion is that I will start from Tom's exposition and adapt it to the framework of $2$-categories. Many thanks for having written that text, Tom!

How do various notions of natural transformation relate to various notions of homotopy in $2Cat$?

2012-01-02T11:26:40Z

In what follows, $2$-categories will be strict, and "$2$-functor" will mean "strict $2$-functor". (Please mention which terminological conventions you are using when answering.) I guess that the answer to my question is rather standard for people working daily with $2Cat$ as a $2$- or $3$-category, and I think I may know some people who could answer my question if I ask them privately. However, it seems to me it could be useful to have an answer to that question on MO for the record (and so as to allow those knowledgeable people to earn much sought-after MO reputation and badges).

It is standard to view natural transformations between functors as categorical homotopies. There has been a question about this viewpoint on MO before. What I would like to know is the relationship between higher natural transformations and homotopies.

The starting point is that, given two categories $A$ and $B$ and two functors $F$ and $G$ from $A$ to $B$, a natural transformation from $F$ to $G$ is the same thing as a functor $\Delta_{1} \times A \to B$ which makes the obvious diagram commutative, where $\Delta_{1}$ is the category associated to the naturally ordered set ${0,1}$.

Now, if we replace $Cat$ by $2Cat$, we have a whole bunch of possible variants. For instance, if $\mathcal{A}$ and $\mathcal{B}$ are two $2$-categories and $F$ and $G$ are two $2$-functors from $\mathcal{A}$ to $\mathcal{B}$, then a lax natural transformation from $F$ to $G$ gives a lax $2$-functor $\Delta_{1} \times \mathcal{A} \to \mathcal{B}$ which makes the obvious diagram commutative. But we could ask whether a lax transformation between lax functors gives such a lax functor too, and whether these two notions are equivalent in this setting, as in the case of $Cat$ (I have just stated one implication only, and just for strict $2$-functors). The question which arise could therefore phrased as:

What are the objects in $2Cat$ analogous to $\Delta_{1} \times A$ in the case of $Cat$, with respect to strict transformations between strict $2$-functors, lax transformations between strict $2$-functors, lax transformations between lax $2$-functors?

I guess this is related to the Gray tensor product, but Gray's style is, woe is me, undecipherable to me. The related $nLab$ page seems to help but not to answer exactly this question (I may well be mistaken).

A related question is the one relating the $2$-categorical viewpoint and the $1$-categorical viewpoint:

What kind of transformations in $2Cat$ gives what kind of homotopies in $2Cat$? (Here, I primarily think to "homotopy" as an arrow from the product of $\Delta_{1}$ with something, but I do not want to restrict to this case if it turns out this is not the right generalization.)

This last question does not ask for the universal objects, but rather how various maps (the "various" refering to the degree of laxness) with various universal objects as domain relate to various (same remark) notions of transformations. I have routinely worked with $2Cat$ from the $1$-categorical viewpoint, but much less from the $2$-categorical viewpoint (let alone the dreaded $3$-categorical viewpoint).

If some people feel comfortable with the general setting of $nCat$, I do not have any objection, although I would mind if it should obfuscate what happens in the special case of $2Cat$.

Many thanks in advance!

Does the category of strict $2$-categories together with Dwyer-Kan equivalences provide a model for $(\infty,1)$-categories?

2011-12-25T09:17:02Z

The question is the title.

In what follows, all $2$-categories and $2$-functors will be strict. Let $2-Cat$ denote the categories whose objects are $2$-categories and whose morphisms are $2$-functors. If $\mathcal{A}$ is a $2$-category, then $\pi_{0}(\mathcal{A})$ is the category whose objects are those of $\mathcal{A}$ and whose $1$-cells are obtained from those of $\mathcal{A}$ by identifying two $1$-cells if they are linked by a zigzag of $2$-cells in $\mathcal{A}$. (In other words, the $Hom$ sets of $\pi_{0}(\mathcal{A})$ are given by the $\pi_{0}$ of the $\underline{Hom}$ categories of $\mathcal{A}$.)

Dwyer-Kan equivalences were defined by Dwyer and Kan in the general setting of simplicial categories. In the realm of $2$-categories, a map $u : \mathcal{A} \to \mathcal{B}$ is a Dwyer-Kan equivalence if the two following conditions are satisfied:

$(i)$ For every objects $X$ and $Y$ of $\mathcal{A}$, the functor $\underline{Hom}_{\mathcal{A}}(X,Y) \to \underline{Hom}_{\mathcal{B}}(u(X),u(Y))$ , induced by $u$, `is a weak equivalence (which means that its nerve is a simplicial weak equivalence or, which is equivalent, that this induced functor belongs to any basic localizer of $Cat$).

$(ii)$ The functor $\pi_{0}(\mathcal{A}) \to \pi_{0}(\mathcal{B})$, induced by $u$, is essentially surjective.

Note that these conditions imply that not only is $\pi_{0}(u)$ essentially surjective, but it is also an equivalence of categories.

Some people have apparently suggested that the localization of $2-Cat$ with respect to the class of Dwyer-Kan equivalences should give, up to equivalence, the category of $(\infty,1)$-categories. However, I have yet to find someone who could point out a proof in the literature or write a proof themselves when asked the question whether this result is more than folkloric belief. Could somebody provide something more concrete? Note that I have no definite clue whether this result is true or not. Arguments against its validity would be welcome without dismay.

Answer by Jonathan Chiche for Find weak equivalences from fibrations and cofibrations

2011-12-22T13:48:55Z

When you have a model structure, the cofibrations and fibrations give you the acyclic cofibrations and acyclic fibrations because of the lifting axioms. Then, the weak equivalences are those arrows which can be factored as an acyclic cofibration followed by an acyclic fibration. (Use the factorization and $2$ out of $3$ axioms.) That is how the class of cofibrations and the class of fibrations give you the whole model structure.

EDIT: While I am at it, it may be worth mentioning that you do not need all the fibrations to recover the model structure once you know the cofibrations. The fibrations whose codomain is the terminal object give you a sufficient data. In other words: a model structure is determined by the cofibrations and fibrant objects. I think this observation is due to Joyal. This is Proposition E.1.10. of his text The Theory of Quasi-Categories and its Applications.

Answer by Jonathan Chiche for The half-life of a theorem, or Arnold's principle at work

2011-05-27T07:14:15Z

I am unsure it fits the OP's requirements but, in connection with Ryan Budney's most-upvoted answer regarding the rediscovery of trapezoidal method, let me recall that Grothendieck spent about three years working in isolation in French provinces developing the Lebesgue theory of integration. It was not before he went to Paris that he was told someone had already done that.

Answer by Jonathan Chiche for What is the homotopy theory of categories?

2011-04-15T08:10:36Z

I am not knowledgeable enough to have much to say I have not writen in my answer to a previous question of yours, and I think that David Roberts's answer (or, rather immodestly, my previous one) provides what you were looking for as regards your first question. Just a few additional small points:

Pursuing Stacks is not a letter. See Tim Porter's comment.

As regards Grothendieck's opinion of Thomason's model structure, I do not know. Actually, I am unsure he knew of Thomason's model structure when writing Pursuing Stacks [EDIT: see Tim Porter's comment below]. What he knew for sure was that the localization of $Cat$ with respect to classical weak equivalences (functors between small categories the nerve of which are simplicial weak equivalences) is equivalent to the classical homotopy category. The first proof is due to Quillen and Illusie "wrote the details" (his words) in his thesis. (And there is a quite simpler proof, by the way.) Model structures crop up in Pursuing Stacks at some point, but I am pretty sure the idea is not developed in the beginning, which is much more concerned with mere models for homotopy types. Here is a citation from Chapter 75: "the notion of asphericity structure — which, together with the closely related notion of contractibility structure, tentatively dealt with before, and the various "test notions" (e.g. test categories and test functors) seems to me the main payoff so far of our effort to come to a grasp of a general formalism of "homotopy models"." (Beware: these asphericity structures are not what Maltsiniotis called "asphericity structures" in his own work.)

Another fact Grothendieck knew was, of course, Quillen's Theorem A. It seems he did not write a detailed proof of the relative version, but he gave a sketch of a toposic proof of it, though, and took it as an axiom for what he called basic localizer.

As for your second question, I do not know, but it seems to me that Grothendieck was not that interested in simplicial sets and thus did not work extensively with them. In a 1991 letter to Thomason, he wrote: " D’autre part, pour moi le "paradis originel" pour l’algèbre topologique n’est nullement la sempiternelle catégorie ∆∧ semi-simpliciale, si utile soit-elle, et encore moins celle des espaces topologiques (qui l’une et l’autre s’envoient dans la 2-catégorie des topos, qui en est comme une enveloppe commune), mais bien la catégorie Cat des petites catégories, vue avec un œil de géomètre par l’ensemble d’intuition, étonnamment riche, provenant des topos. En effet, les topos ayant comme catégories des faisceaux d’ensembles les C∧ , avec C dans Cat, sont de loin les plus simples des topos connus, et c’est pour l’avoir senti que j’insiste tant sur l’exemple de ces topos ("catégoriques") dans SGA 4 IV". (See here.)

To conclude, let me mention that, if one takes Grothendieck's viewpoint of homotopical algebra, there should exist not only a homotopy theory of categories, but a homotopy theory of $n$-categories. In this respect, there should be a "relative Theorem A" for every $n$, which should allow one to define a workable notion of "basic $n$-localizer". (Actually, this is already done for $n=2$: see this paper by Bullejos and Cegarra for Theorem A.) And then one should work out a theory of test $n$-categories, whose $(n-1)-Cat$-valued presheaves should be models for homotopy types, and so on. To sum up, what Grothendieck wanted to do amounts to giving new foundations for homotopical algebra, and this is still a work in progress.

David Roberts gives the two most useful available references in his answer. If you want to read Grothendieck's words (and in English), just wait for the upcoming annotated version of Pursuing Stacks.

Answer by Jonathan Chiche for Is there a high-concept explanation for why "simplicial" leads to "homotopy-theoretic"?

2011-03-15T08:47:12Z

There are several people here much more qualified to speak about that, so I shall just give you some pointers now. One of the questions Grothendieck tried to answer when writing "Pursuing Stacks" was — I don't know how he put it, though — "what are the properties of the simplicial category which make it so useful in homotopy theory?" That is where the theory of test categories stems from. As Georges Maltsiniotis puts it: "Le slogan de Grothendieck est que toute catégorie test est aussi “bonne” que celle des ensembles simpliciaux pour “faire de l’homotopie”." Which means "Grothendieck's motto is that any test category is as "good" as the category of simplicial sets to "make homotopy theory"." The theory was further developed by Denis-Charles Cisinski. The two books to read on this subject are:

Maltsiniotis's "La Théorie de l'homotopie de Grothendieck" ("Grothendieck's Homotopy Theory"), the introduction of which is remarkably well-written:
http://www.math.jussieu.fr/~maltsin/ps/prstnew.pdf

and

Cisinski's (augmented version of his) thesis "Les Préfaisceaux comme modèles des types d'homotopie" ("Presheaves as Models for Homotopy Types"): http://www.math.univ-toulouse.fr/~dcisinsk/ast.pdf

Both are available in SMF's collection Astérisque.

I shall give you more details if nobody else shows up to explain the yoga (I myself have but a smattering of it).

EDIT: Well, here are some details. You are asking: "What is so wonderful about Δ that allows a model structure (and one, moreover, Quillen equivalent to topological spaces) appear?" The shortest answer would be: "$\Delta$ is a test category". Let's try to see what it means. (I am feeling a bit guilty, for what follows is essentially a rephrasing, with the same notations, of some parts of Maltsiniotis's crystal-clear introduction to his book. I hope it will at least benefit those who cannot read French. Please note that Maltsiniotis's book is based on material written by Grothendieck in "Pursuing Stacks" almost thirty years ago.)

The starting point of the theory of test categorie is similar to your question. Namely, Grothendieck seeks to find all the couples $(M, W)$ where $M$ is a category and $W \subseteq Ar(M)$ such that the localized category $W^{-1}M$ be equivalent to the homotopy category $Hot$, and such that $W$ is natural in some sense (with respect to the structure of the underlying category). Given the difficulty to answer such general a question, Grothendieck then requires of $M$ to be a presheaf category on a small category $A$. Adding another slight condition on the small category $A$ (requiring that the "nerve functor" $i_{A}^{*} : Cat \to \widehat{A}$, $C \to (a \mapsto Hom_{Cat}(A/a, C))$, send weak equivalences to weak equivalences, where weak equivalences of $Cat$ are those functors the classical nerve of which are simplicial weak equivalences, and weak equivalences in the presheaf category $\widehat{A}$ are those morphisms sent to weak equivalences of $Cat$ by the functor $A/?$), he is lead to define the notion of weak test category. One of the properties of such a category $A$ is that the localization of its presheaf category by weak equivalences is equivalent to the homotopy category $Hot$. Of course, the simplicial category is a test category. But it is even better that that. It is a strict test category, which implies (by definition) for instance that cartesian product reflects the product of homotopy types. This theory shows, by the way, that the cubical category differs from the simplicial category in this respect: indeed, the cubical category is not a strict test category (but it is a test category, which of course lies somewhere between being weak test and being strict test). You might think that, since the cubical category is not a strict test category, strict test categories ought to be pretty scarce. In fact, there are plenty of them. For instance, every full subcategory of $Cat$ the objects of which are non-empty, and which is stable under finite products, and one object of which has at least two objects (possibly isomorphic) is a strict test category. There are results allowing one to check that a given category is a (weak, local, strict…) test category, which I will not state here. Just one example: Joyal's category $\Theta$ (related to infinity stuff) is a test category (this was proved by Cisinski/Maltsiniotis and Ara).

Actually, there is more than that in the theory. You can ask what are the formal properties of weak equivalences of $Cat$ that make the theory works so well. That is what Grothendieck answered by defining basic localizers. Indeed, what you need is just a class $W$ of functors between small categories such that: $W$ is weakly saturated (which means it contains identities, it satisfies a two out of three axiom, and if $i$ has a retraction such that $ir$ is in $W$, then $i$ (and thus $r$) is in $W$) ; if $A$ is a small category which has a terminal object, then $A \to e$ is in $W$ ($e$ stands for the point category) ; and $W$ satisfies the relative version of Quillen's Theorem A. That is all you need to develop the theory of test categories. Grothendieck then proceeds to rewrite all the theory with respect to an arbitrary basic localizer replacing $\mathcal{W}_{\infty}$, the classical weak equivalences of $Cat$.Therefore, for every basic localizer $W$, there are notions of $W$-weak test category, $W$-local test category, $W$-test category, $W$-strict test category and so on. Truncated homotopy types provide instances of basic localizers $\mathcal{W}_{n}$ for every $n \geq 0$, but there are many others.

And here is a theorem: for every basic localizer $W$, for every $W$-test category $A$, there is a closed model category structure on the category of presheaves on $A$, the weak equivalences of which are those defined above (so that, in particular, the localized category is equivalent to the localized category $W^{-1}Cat$) and the cofibrations of which are the monomorphisms. In fact, you have to make a slight set theoretic assumption for this result to hold (namely, that the basic localizer is accessible, that is, it is the smallest one containing some set of arrows). It was conjectured by Grothendieck and proved by Cisinski.

OK, now it might still be unclear as to what are the advantages of this theory. One of them is that you can work with other basic localizers than the classical one (the $W_{\infty}$ of above). Classical weak equivalences are related to Artin-Mazur equivalences in slice presheaves toposes, and these can be replaced, for instance, by any other topos morphisms defined by cohomological properties. (See the first paragraph of page 12 of Maltsiniotis's book, for instance.)

There are much more stuff in Grothendieck's homotopy theory, but I shall limit myself to that now.

By the way, there has been a very nice expository talk (in French) by Maltsiniotis on Grothendieck's 1980's work at IHES two years ago:
http://www.dailymotion.com/video/x8jsnw_colloque-grothendieck-georges-malts_tech.

EDIT: I just added some details and thought I could elaborate on two points of Jacob Lurie's answer as well in the language of Grothendieck's homotopy theory (which I of course do not claim to be better). When he states that the (op)-siftedness of the simplicial category guarantees "a nice connection between the homotopy theory of simplicial sets and the homotopy theory of bisimplicial sets", I guess the key result he is alluding to is the classical "bisimplicial lemma", which states that, if $f : X \to Y$ is a bisimplicial morphism such that $f_{n,.} : X_{n,.} \to Y_{n,.}$ is a simplicial weak equivalence for every $n \geq 0$, then $\delta^{\ast}(f):\delta^{\ast}X \to \delta^{\ast}Y$ is a simplicial weak equivalence. Here, $\delta : \Delta \to \Delta \times \Delta$ stands for the diagonal functor, and $\delta^{\ast}$ for the induced functor which send a bisimplicial set $X$ to the simplicial set $n \mapsto X_{n,n}$. I would like to point out that a similar result holds for every totally aspherical category, that is, a small category $A$ such that the functor $A \to e$ is a weak equivalence (which means that it belongs to the basic localizer we are considering) and such that (one among many equivalent properties) the diagonal functor $A \to A \times A$ is aspherical (which means that for every $(a_{1}, a_{2}) \in A \times A$, the comma category $\delta \downarrow (a_{1}, a_{2})$ is aspherical). For such a category $A$, whenever $f$ is a morphism in the category of presheaves $\widehat{A \times A}$ such that $f_{a,.}$ is a weak equivalence for all $a \in A$, then $\delta^{\ast}f$ is a weak equivalence (in the category of presheaves, see above). The simplicial category $\Delta$ is $W_{\infty}$-totally aspherical, a (non-trivial) fact from which one can deduce the "bisimplicial lemma". The siftedness has to do with the $W_{0}$-total asphericity, therefore I was puzzled as to how to deduce the "bisimplicial lemma" from it (one needs $W_{\infty}$ as basic localizer). It seems Jacob Lurie is tacitly taking the $(\infty,1)$-categorical viewpoint, which makes the two properties equivalent. (Thanks to Georges Maltsiniotis for poiting that to me.)

As to Dold-Kan correspondence, I asked Maltsiniotis if a similar result holds with other Grothendieck test categories, and the answer is that there is no such result in general, but there is already a conjecture in "Pursuing Stacks" regarding an analogous correspondence for any strict test category.

I am not sure many people wanted to read all that but I thought I would share what I knew since this stuff is not written down in any currently available text.

Has the mathematical content of Grothendieck's "Récoltes et Semailles" been used?

2010-12-08T10:22:35Z

This question is partly motivated by this one.

Motivation

Grothendieck's "Récoltes et Semailles" has been cited on various occasions on this forum. See for instance the answers to this question or this one. However, these citations reflect only one aspect of "Récoltes et Semailles", namely the nontechnical reflexion about Mathematics and mathematical activity. Putting aside the wonderful "Clef du Yin et du Yang", whis is a great reading almost unrelated to Mathematics, I remember reading in "Récoltes et Semailles" a bunch of technical mathematical reflexions, almost all of which were above my head due to my having but a smattering of algebraic geometry. However, I recall for instance reading Grothendieck's opinion that standard conjectures were false, and claiming he had in mind a few related conjectures (which he doesn't state precisely) which might turn out to be the right ones. I still don't even know what the standard conjectures state and thus didn't understand anything, but I know many people are working hard to prove these conjectures. I've thus often wondered what was the value of Grothendieck's mathematical statements (which are not limited to standard conjectures) in "Récoltes et Semailles".

The questions I'd like to ask here are the following:

Have the mathematical parts of "Récoltes et Semailles" proved influential? If so, is there any written evidence of it, or any account of the development of the mathematical ideas that Grothendieck has expressed in this text? If the answer to the first question is negative, what are the difficulties involved in implementing Grothendieck's ideas?

Idle thoughts

In the latter case, I could come up with some possible explanations:
1. Those who could have developed and spread these ideas didn't read "Récoltes et Semailles" seriously and thus nobody was aware of their existence.
2. Those people took the mathematical content seriously but it was beyond anyone's reach to understand what Grothendieck was trying to get at because of the idiosyncratic writing style.
Should one of these two suppositions be backed by evidence, I'd appreciate a factual answer.
3. The ideas were already outdated or have been proven wrong.
If this is the case, I'd appreciate a reference.

Epanorthosis

Given that "Pursuing Stacks" and "Les Dérivateurs" were written approximately in 1983 and 1990 respectively and have proved influential (see Maltsiniotis's page for the latter text, somewhat less known), I would be surprised should Grothendieck's mathematical ideas expressed around 1985 be worthless.

Comment by Jonathan Chiche

2013-04-06T15:58:37Z

I understand this sentence as « I have received no answer on Facebook ». I would downvote the question if I was sure I am not mistaken.

Comment by Jonathan Chiche

2013-03-15T06:44:28Z

There seems to be some misunderstanding, perhaps because of linguistic issues. According to this article (which I was aware of), Grothendieck's papers have not been declared "trésor national" yet. This is (or was) only a project that some people have (or had) in mind. As a side remark, Guy Debord's papers have been declared "trésor national". Both stories are interesting.

Comment by Jonathan Chiche

2013-03-04T20:15:14Z

I have read the title as "The role of the agence nationale de la recherche in modern topology".

Comment by Jonathan Chiche

2013-02-13T22:29:58Z

But you did not mention accusations of plagiarism in your answer... I was not aware of that. Thanks.

Comment by Jonathan Chiche

2013-02-12T09:11:12Z

Found: home.broadpark.no/~emeyn/tl/radio2.html. "I took the name of Lobachevsky, only for prosodic reasons, just because it fit". (With an emphasis on "only".)

Comment by Jonathan Chiche

2013-02-12T09:04:13Z

If my memory serves me right, Lehrer made this choice for prosodic reasons only. Or euphonic ones. I mean, it sounds better than other names and fits well. I recall Lehrer talking about that in an interview, to which one could probably find a link on a Youtube channel devoted to him.

Comment by Jonathan Chiche

2012-12-11T09:51:31Z

(ctd) written anything interesting from a mathematical point of view at this period. Even Quillen did not answer to the letter! It is also true that, from what I have been told, it was almost impossible to "sell" the issues raised by Grothendieck at the time. For instance, Bénabou had discussed the question of higher analogs of bicategories (that is, weak n-categories) with Grothendieck around 1966 but he told me that basically he thought he could not have been "forgiven" to work on such a topic, and that only Grothendieck could be forgiven. I am unsure he has been.

Comment by Jonathan Chiche

2012-12-11T09:41:50Z

David, thanks for your comments. I hope you did not understand I was claiming nobody had read "Pursuing Stacks" and I am glad you have. But how many precise references to this text could one point out in the literature? It is true that some people present (I am not qualified to judge whether they are right or not) their work as solutions to questions raised by Grothendieck, but what has been written about the solutions brought by Grothendieck himself? This is something which I find very sad, but for many years most people seem to have been thinking that Grothendieck could not have (ctd)

Comment by Jonathan Chiche

2012-11-10T18:19:53Z

Thanks. I have not looked at the details, mainly because I was in fact not asking for a proof, but rather a reference. I knew that the classical proof that, say, arbitrary multiplication in a monoid is well-defined, could be adapted to show the coherence result here. The fact which I find disturbing is that there does not seem to be any published text containing both a rigorous statement and a rigorous proof.

Comment by Jonathan Chiche

2012-11-07T19:50:59Z

Tom: thanks for your explanation, it makes sense. However, I still find "hypothesis" a somewhat unfortunate choice, albeit justified from the point of view you describe (but I am unsure what was needed to make the statement precise at that time), because it seems to me too many people now consider it to be a kind of axiom. More generally, Grothendieck's text is probably considered much more speculative than it really is.

Comment by Jonathan Chiche

2012-11-07T18:11:54Z

By the way, there might be an interesting question lurking here. (But, of course, this question should be drastically rewritten.)

Comment by Jonathan Chiche

2012-11-07T18:10:30Z

I think Tom is right. This should be clear from Grothendieck's letter to Quillen. But why on earth do people say "Homotopy Hypothesis" and not "Homotopy Conjecture"? In Grothendieck's text there is a perfectly valid definition of \infty-groupoid, and the equivalence of categories is then a conjecture, not something to be seen as right or wrong depending on your personal beliefs. (Dimitri Ara's thesis and papers may (I am not an expert) be the best place to learn that.)

Comment by Jonathan Chiche

2012-11-01T11:50:16Z

Well, I know what the axioms are, but part of my point is that some of these axioms are missing in your question. For instance, you don't mention the fact that lax functors induce functors between the categories of 1-cells. I don't think this condition is automatically satisfied as soon as the ones you mention are, even in the realm of (strict) 2-categories.

Comment by Jonathan Chiche

2012-10-22T07:31:53Z

Thanks! I am noy sure this is what I am looking for but it may be helpful.

Comment by Jonathan Chiche

2012-10-22T07:30:43Z

Thanks to those who expressed their opinion by upvoting Andrej Bauer's comment. Special thanks to Mike, who made me aware that I was not the only one not to get the LaTeX parsed in the main questions list. I have now edited the title accordingly.