## May 11, 2009

### Journal Club – Geometric Infinity-Function Theory – Week 3

#### Posted by Urs Schreiber

This week in our Journal Club on [[geometric $\infty$-function theory]] Bruce Bartlett talks about section 3 of “Integral Transforms”: perfect stacks.

Week 1: Alex Hoffnung on Introduction

Week 2, myself on Preliminaries

See here for our further schedule. We are still looking for volunteers who’d like to chat about section 5 and 6.

Journal Club entry by Bruce Bartlett

Well it’s my turn to give a write-up in the Journal Club. The topic for this week is perfect stacks, and it concerns Section 3 of David Ben-Zvi, John Francis and David Nadler’s paper Integral transforms and Drinfeld centers in derived algebraic geometry.

As I understand it, we need the concept of perfect stacks because they represent the most general context in which the important theorems of geometric infinity-function theory hold — namely, that algebraic and geometric operations on their categories of sheaves are nicely compatible.

This is even more succinctly put in the heading area of the paper:

Compact objects are as necessary to this subject as air to breathe.

That, by the way, is apparantly a statement made by R.W. Thomson, to A. Neeman, to the authors via the book Triangulated categories, to me via the paper we are studying, and now to you, dear reader!

Now I’m pretty shoddy on some of the more homological/algebro-geometric aspects that I need to understand this stuff properly, being more accustomed to working with down-to-earth things such as smooth manifolds and differential forms, as opposed to spectra of rings and such like. Urs and Alex perhaps come from a similar viewpoint. Chris might know this stuff quite well. Anyhow, I’m going to try and keep Urs’s $\infty$-stacky incarnation of smooth geometry via generalized smooth spaces and $\omega$-groupoids on the side, and ask questions as we go along about how “perfectness” translates into this context, if indeed it does.

As I understand it, a derived stack $X$ is roughly speaking something which assigns a $\infty$-groupoid $X(U)$ to each ‘space’ $U$

(1)$U \mapsto X(U)$

in a manner which is compatible with gluing. In other words, a derived stack is an ‘$\infty$-sheaf’. In the case of this paper, we fix a ‘commutative derived ring’ $k$, and then the category of ‘spaces’ in question is the opposite of the $\infty$-category of derived $k$-algebras.

The concrete application closest to my sphere of understanding is the case where $k=\mathbb{C}$, so that the category of spaces’ is the opposite of the category of connective commutative differential graded algebras over $\mathbb{C}$.

To go over into Urs’s framework, I think the category of ‘spaces’ is just the category $Man$ of good old smooth manifolds, while the concept of $\infty$-groupoid translates into ‘globular strict $\omega$-groupoid’.

So in Urs’s setup, a derived stack’ is nothing but a smooth $\omega$-groupoid.

Question 1. Is this correct?

Okay, now suppose we’ve got one of these derived stacks $X$. The natural notion of a function on one of these guys is the $\infty$-category of quasi-coherent sheaves over $X$, written $QC(X)$. The way to define $QC(X)$ is as follows. Any stack $X$ can be expressed as a gluing or colimit of affine derived schemes:

(2)$X \simeq colim_{U \in Aff_{/X}} U.$

I’m not exactly sure what an affine derived scheme is, but in Urs’s setup I think this means that any generalized smooth space can be expressed as a colimit of (the stacks represented by) smooth manifolds:

(3)$X \simeq colim_{U \in Man_{/X}} U.$

Come to think of it, I’m not quite sure what that means either, but I know it has something to do with the ‘coYoneda lemma’ which I stumbled onto on the nLab.

Anyhow, an affine thingy is something of the form $U = Spec A$ for some algebra thingy $A$, so we can define $QC(U)$ for affine thingys to be to be the $\infty$-category of $A$-modules. And then we can define $QC(X)$ for a general stack $X$ as the appropriate gluing together of these module categories:

(4)$QC(X) = lim_{U \in Aff_{/X}} QC(U).$

Question 2. I’m actually a bit uncomfortable with this, since it’s not a concrete ‘definition’ in the classical sense of the word, since the original expression of $X$ as a colimit wasn’t canonical, and neither is the limit above. I know these sorts of definitions are rife in homotopy theory though, so I guess I must just accept it? I would be happier if there was a nice canonical concrete expression.

Question 3. In Urs’s picture, does the concept of $QC(X)$ make sense? Is it the $\infty$-category of vector-bundles on $X$ (possibly equipped with connection) Or is it more of a ‘local-system’ thing?

Ok, so here’s what it means for a derived stack to be perfect. We want $QC(X)$ to be ‘generated’ by some smaller subcategory consisting of ‘finite objects’. There are two candidates for what ‘generated’ might mean (we’ll get to this) and there are three candidates for what ‘finite objects’ might mean.

The good news is that we don’t have to know a lot about quasicoherent sheaves to discuss this matter.

The three candidates for what ‘finite object’ might mean are as follows.

• An object $M \in QC(X)$ is perfect if its restriction $f^*M$ to any affine chart $f : U \rightarrow X$ is perfect, in the sense that it lies in the smallest subcategory of $QC(U)$ containing $A = Spec(U)$ and which is closed under finite colimits and retracts.
• An object $M \in QC(X)$ is compact if $Hom(M, \cdot)$ commutes with all colimits (equivalently, with all coproducts).
• An object $M \in QC(X)$ is strongly dualizable (or dualizable for short) if it has a dual $M^\vee$ in the categorical sense — that is, if there exists an object $M^\vee$ together with unit and trace maps $u : 1 \rightarrow M^\vee \otimes M$ and $\tau : M \otimes M^\vee \rightarrow 1$ satisfying the snake diagrams.
The cool thing is that these three definitions are clearly quite natural, and they ask questions of the geometry of $X$, the categorical structure of $QC(X)$, and the monoidal structure of $QC(X)$ respectively.

I wish I could get some more familiarity with how these three concepts work out in specific examples. From the nLab, I see that the concept of a ‘compact’ object can be applied to any category, except in general one should ask that $Hom(M, \cdot)$ only commutes with all filtered colimits. There’s a nice example: a topological space $X \in Top$ is compact if and only if the hom-functor

(5)$\Hom(X, \cdot) : Top \rightarrow Top$

commutes with all filtered colimits. I’d like to understand this example!

I wish I could understand this concept of filtered colimit better. I need to go back to point-set topology and understand what a ‘filter’ means. I wish John would write an expository article explaining filters (in topological spaces), and how they generalize to filtered colimits.

Anyhow, the interesting fact is that these three notions (perfecness, compactness, dualizable) all coincide in the affine situation. Moreover, ‘strongly dualizable’ and ‘perfect’ are apparantly local properties, so therefore they always coincide. The only problem-child is ‘compact object’, which is not a local property and hence might not glue nicely. More on that later.

Having discussed the various candidates for the notion of ‘finite object’, we must still discuss what we are going to mean when we say that they ‘generate’ our category $QC(X)$. The two candidates are:

• $QC(X)$ is the inductive limit of the category of finite objects (basically, this says that $QC(X)$ is the formal completion of the category of finite objects).
• $QC(X)$ is compactly generated by the finite objects — that is, there is a set of finite objects $C_i \in C$ such that if $Hom(C_i, M) \cong 0$ for all $i$, then $M \cong 0$.

The choice made in this paper is as follows.

Definition. A derived stack $X$ is said to be perfect if it has affine diagonal and if the $\infty$-category $QC(X)$ is the inductive limit

(6)$QC(X) \simeq Ind Perf (X)$

of the full $\infty$-subcategory $Perf(X)$ of perfect complexes. A morphism $X \rightarrow Y$ is said to be perfect if its fibers $X \times_Y U$ over affines $U \rightarrow Y$ are perfect.

The main point is that we could have used any of the other notions, because of the following nice fact.

Proposition. For a derived stack $X$ with affine diagonal, the following are equivalent:

• $X$ is perfect.
• $QC(X)$ is compactly generated, and its compact and dualizable objects coincide.

The proof of this apparantly comes down to checking the following lemma:

Lemma. If $N$ is a retract of a dualizable object $M$, then $N$ is also dualizable.

I tried to prove this using string diagrams, but I got stuck. I think it’s because I’m not incorporating the closed structure (ie. the internal hom).

Question 4. Give a nice string diagram proof of this statement.

So let’s summarize where we are (we’re only at the end of Section 3.1!):

• We’ve defined the derived category $QC(X)$ of quasicoherent sheaves on a derived stack $X$.
• We’ve said what it means for $QC(X)$ to be perfect. This boiled down to making one of a number of different natural choices. Happily, they turn out all to be equivalent, precisely when $QC(X)$ is perfect!
• The good news for n-category cafe denizens is that all of these choices appear to be reasonably ‘elementary’ and don’t require sky-high formalism. This appears to go for the proofs too. Thus the concept of ‘perfectness’ has something fundamental and important to say about monoidal-ish categories (see Question 4 above, as well as the fact that being ‘closed under retracts’ ties in very nicely with the 2-Hilbert space formalism). Let’s try to get to grips with it.

The remaining two sections deal with base change and the projection formula for perfect morphisms (section 3.2), and that many common classes of stacks are perfect (section 3.3). I think one can only appreciate these sections properly once one has been working with schemes and algebro-geometric stacks for a while… then when one reads the results, one will say “oh, that’s nice, because that particular property was always a problem in the so-and-so approach”. Sadly I am not in that boat.

Posted at May 11, 2009 11:04 PM UTC

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### Re: Journal Club – Geometric Infinity-Function Theory – Week 3

Thanks, Bruce!

I was busy all day, now it’s late at night and I’ll be busy half of next day at least, but then I’ll try to respond to you here.

Just one quick remark on question 1:

I am not sure what “my setup” refers to exactly, but the fllowing statement about derived stacks is the thing to keep in mind (I mean, you know this, we talked about this at length back then):

a stack on a site $S$ is a groupoid “modeled on $S$”.

a stack on $S = Diff$ in particular is a smooth groupoid.

a stack on $S = Alg^{op}$ in particulat is an algebraic groupoid.

Now, there are clearly two ways here to increase the categorical dimension: on the domain and on the codomain:

an $\infty$-stack on $S$ is an $\infty$-groupoid modeled on $S$. So

an $\infty$-stack on $S = Diff$ in particular is a smooth $\infty$-groupoid.

an $\infty$-stack on $S = Alg^{op}$ in particular is an algebraic $\infty$-groupoid.

But there is no good point in increasing just the categorical codomain and not the domain. So we could just as well consider $\infty$-groupoids which are probeable not just by manifolds, but, say, by cosimplicial manifolds $S = Diff^{Delta}$. Since cosimplicial manifolds themselves form a higher category, we call an $\infty$-stack on a higher categorical site such as $Diff^{\Delta}$ a derived stack, just so that it sounds fancy and no outsider knows what we are talking about!

So these would be still “smooth $\infty$-groupoids” only that now the notion of what “smooth” means is much more flexible.

Similarly, we can consider simplicial algebras or simplicial rings dualized and $\infty$-stacks on them. That yields algebraic derived stacks. And they are still “algebraic $\infty$-groupoids” only that now the notion of what “algebraic” means is much more flexible.

None of this has to do directly with strict versus weak $\infty$-groupoids. The only point about strict $\infty$-groupoids is that often they are sufficient and more easily worked with than the fully general $\infty$-groupoids. I’ll provide some concrete examples later.

Posted by: Urs Schreiber on May 11, 2009 11:41 PM | Permalink | Reply to this

### Re: Journal Club – Geometric Infinity-Function Theory – Week 3

Thanks for another great report!

Regarding derived stacks: as Alex summarized in his post, the idea is

1) stacks and higher stacks arise from quotients (or more general colimits) on schemes

2) derived stacks arise from fiber products (or more general limits) on schemes and stacks

i.e. the reason to change the domain of functors is that intersections of manifolds, and more general fiber products - for example inverse images of nonregular values - can be badly behaved.
So we want to keep track of this, and this can be done by changing the domain of our functors from manifolds to something like COsimplicial manifolds.
(cosimplicial schemes are opposite to simplicial rings..)

A less abstract model for this source category is differential graded supermanifolds.. algebraically, one thinks of varieties with a structure sheaf not just of commutative rings but of (negatively graded) commutative dgas.
This is where constructions like virtual fundamental classes, and other corrections for nontransversality and nonregularity, naturally live..

So a derived stack can be thought of as something obtained by taking quotients of dg manifolds by groups or more generally (higher) groupoids..

Posted by: David Ben-Zvi on May 12, 2009 4:17 AM | Permalink | Reply to this

### Re: Journal Club – Geometric Infinity-Function Theory – Week 3

So a derived stack can be thought of as something obtained by taking quotients of dg manifolds by groups or more generally (higher) groupoids..

Ok, thanks a lot. That gives me something to pin it on. I never absorbed properly the fact that derived stacks involved a change of range as well as domain of the functors, for the reasons you described, though indeed I should have from Alex’s post.

I’ve been reading through the discussion about what “character TFT” would mean in a smooth context. Seems quite tough. But I am happy that at least I now have a concrete smooth example of a derived stack $X$: an $\omega$-groupoid internal to differential graded supermanifolds. (Not that I am very comfortable with supermanifolds, etc., but it’s at least something I’ve tried to study before and which is close to my interests, etc.)

Now just need some kind of rough analogue $QC(X)$, the derived category of quasicoherent sheaves on $X$, in the case where $X$ is one of these ‘smooth super $\omega$-groupoids’.

So we could just as well consider $\infty$-groupoids which are probeable not just by manifolds, but, say, by cosimplicial manifolds $S=Diff^{Delta}$.

Ok, thanks for reminding me of this. If we want to equip our probing category with higher morphisms, could we also just set $S = smooth \omega-groupoids$? Is that ‘wrong’ because $S = Diff^{Delta}$ has higher cells which are not all invertible? In that case, how about $S = smooth \omega categories$?

Posted by: Bruce Bartlett on May 12, 2009 11:05 AM | Permalink | Reply to this

### Re: Journal Club – Geometric Infinity-Function Theory – Week 3

David wrote, on the need for $\infty$-stacks whose domain is not a 1-category but a higher category (aka “derived stacks”):

2) derived stacks arise from fiber products (or more general limits) on schemes and stacks

i.e. the reason to change the domain of functors is that intersections of manifolds, and more general fiber products - for example inverse images of nonregular values - can be badly behaved. So we want to keep track of this, and this can be done by changing the domain of our functors from manifolds to something like COsimplicial manifolds. (cosimplicial schemes are opposite to simplicial rings..)

[…]

This is where constructions like virtual fundamental classes, and other corrections for nontransversality and nonregularity, naturally live..

Just for the record, for those who need to see more background on this particular point:

A careful motivation of the need of categorified function rings from this perspective of “fiber products that correctly keep track of how they arose” is, guess where, in the introductory section of

- Jacob Lurie, Structured spaces

see pages 2 to 4, especially 3 (just to emphasize that it is a reading load of order of magnitude $10^0$ pages :-)

Similar comments with an eye more towards [[derived smooth manifolds]] are in section 0.1 of

* David Spivak, Quasi-smooth derived manifolds.

It seems to me that a quick way to see what the point here is is to remember what the statement is for the codomain side of the story:

let $X$ be a manifold with non-free action by a group $G$. Then

- $X/G$ may not exist as an object in $Diff$;

- $X/G$ exists as an object in $[Diff^{op},Set]$ but that will not have the expected cohomology;

- $X/G$ exists as an object in $[Diff^{op},Grpd]$ – where it is the smooth [[action groupoid]] – and that will have the right cohomology: namely the $G$-equivariant cohomology of $X$.

So the thing is: while passing to sheaves is sufficient for having all colimits, these colimits may not be the right ones in that they fail to “remember how they arose”. Going further to $\infty$-sheaves takes care of that.

(This is, by the way, precisely the issue that we discussed in the context of higher Lie theory here.)

The story about derived $\infty$-stacks, i.e. the need for the higher categorical domain – is the dual story to that.

I have now added comments along these lines to the entry on [[derived stack]].

Hopefully somebody will feel sufficiently appalled by the insufficiency of this entry to start helping out…

Posted by: Urs Schreiber on May 12, 2009 8:18 PM | Permalink | Reply to this

### Re: Journal Club – Geometric Infinity-Function Theory – Week 3

Ok, so now we have a concept of a ‘derived smooth manifold’, and a beautiful PhD thesis to explain it (I missed this when it was discussed here.). We even have thoughts by Urs that a derived smooth manifold is like a ‘smooth $\infty$-category’.

In lieu of David’s (B-Z not S) comment above above about differential graded supermanifolds, my question is: what is the relationship between derived smooth manifolds and differential graded supermanifolds? My understanding is that they play the same conceptual role (models for smooth $\infty$-categories) so they should be related.

Posted by: Bruce Bartlett on May 13, 2009 10:00 PM | Permalink | Reply to this

### Re: Journal Club – Geometric Infinity-Function Theory – Week 3

COsimplicial manifolds

Ah, right, thanks. I have corrected this in my comment above.

A less abstract model for this source category is differential graded supermanifolds

Ah, of course, I should add a discussion to the $n$Lab entry on [[derived stack]] relating it to [[NQ-supermanifold]]

If only I had a bit more time…

Posted by: Urs Schreiber on May 12, 2009 8:35 AM | Permalink | Reply to this

### Re: Journal Club – Geometric Infinity-Function Theory – Week 3

I’m not exactly sure what an affine derived scheme is

That’s just mighty words for the most trivial thing here:

A scheme is a sheaf with nice properties on the site [[$CRing^{op}$]] of objects dual to commutative rings. Generalized spaces modeled on spaces dual to commutative rings.

An affine scheme is a generalized space which happens to be not generalized: its just an object of $CRing^{op}$ regarded as a generalized space, i.e. the sheaf [[represented]] by that object.

Maybe think of this in terms of smooth spaces. Let the category of test objects be $CartesianSpaces$, the full subcategory in $Diff$ on manifolds of the form $\mathbb{R}^n$. These are the “affine” smooth manifolds.

A general sheaf on $CartesianSpaces$ is a pretty general kind of smooth space, or “smooth scheme”, if you insist (compare [[structured generalized space]] for more on that). But also each $\mathbb{R}^n$ defines such a sheaf. These are the “affine” generalized spaces.

affine space = test space

If we are now talking about derived stacks, then our site is no longer just $CRing^{op}$, but $SimpCRing^{op}$, the simplicially enriched category of [[simplicial objects]] in $CRing$, to be thought of under the [[Dold-Kan correspondence]] as [[differential graded algebras]] (if we are talking about rings under some field).

Whatever the generalized spaces modeled on these (the derived $\infty$-stacks) are, the affine derived $\infty$-stacks are simply the duals to the differential graded algebras themselves. So these are just the ordinary [[NQ-super”manifold”s]] themselves.

Posted by: Urs Schreiber on May 12, 2009 1:51 PM | Permalink | Reply to this

### Re: Journal Club – Geometric Infinity-Function Theory – Week 3

I wrote:

affine space = test space

David Spivak gives a precise formalization of this statement in def 1.1.2 of his thesis. See [[category of local models]].

Posted by: Urs Schreiber on May 12, 2009 10:29 PM | Permalink | Reply to this

### Re: Journal Club – Geometric Infinity-Function Theory – Week 3

A derived stack X is said to be perfect if it has affine diagonal and if the $\infty$-category $QC(X)$ is the inductive limit

(1)$QC(X) \simeq Ind Perf(X)$

Not quite. The notation $Ind C$ is not read “inductive limit”=”colimit” of $X$. There is no notion of colimit over an object.

The notation $Ind C$ or $Ind-C$ or $ind-C$ or the like is the category of [[ind-objects]] of $C$.

These are all the those things which one would get as small finltered colimits in $C$, if these colimits existed in $C$.

Here, more precisely, $Ind C$ is the $(\infty,1)$-category of [[ind-objects in an $(\infty,1)$-category]].

By the way, the whole formalism later on is built in such a way that colimits in $QC(X)$ litearlly play the role of categorified sums in the sense of the categorified linear algebra which we are doing here. (For instance the tensor product $QC(X)\otimes QC(Y)$ is literally built as the universal object for “bilinear” namely colimit preserving $(\infty,1)$-functors, exactly the way the ordinary tensor product of, say, ordinary functions, is built ).

So one should think of

$Perf(X) = something like bounded functions on X$

$Ind Perf(X) = something like formal series of bounded functions on X$

I think. (But maybe there is a better interpretation than “bounded $\infty$-functions” for the things in $Perf(X)$).

Posted by: Urs Schreiber on May 12, 2009 2:12 PM | Permalink | Reply to this

### Re: Journal Club – Geometric Infinity-Function Theory – Week 3

Not quite.

Hey, that’s word-for-word from the original article! I think they meant “inductive limit” in the sense that “Ind $C$” is read as the category of ind-objects of $C$, as you say. Maybe it should be called “completion” or something, I don’t know. Anyhow, I appreciate the intuition about what Perf($X$) and Ind Perf ($X$) are about.

Posted by: Bruce Bartlett on May 13, 2009 10:13 PM | Permalink | Reply to this

### Re: Journal Club – Geometric Infinity-Function Theory – Week 3

Hey, that’s word-for-word from the original article!

Oops. Okay, hm, is that standard terminology? On page 3 at least it says “inductive limit or ind-category”.

Posted by: Urs Schreiber on May 13, 2009 10:24 PM | Permalink | Reply to this

### Re: Journal Club – Geometric Infinity-Function Theory – Week 3

On page 3 at least it says “inductive limit or ind-category”.

I'd be inclined to read that as ‘inductive limit category’ or ‘ind-category’.

Posted by: Toby Bartels on May 14, 2009 4:55 AM | Permalink | Reply to this

### Re: Journal Club – Geometric Infinity-Function Theory – Week 3

On page 3 at least it says “inductive limit or ind-category”.

I’d be inclined to read that as ‘inductive limit category’ or ‘ind-category’.

Right, that’s what I thought, too. But later on it is explicitly not used that way.

Posted by: Urs Schreiber on May 14, 2009 9:05 AM | Permalink | Reply to this

### Re: Journal Club – Geometric Infinity-Function Theory – Week 3

I have a question on the following observation:

it strikes me that what we started calling part 1 of the fundamental theorem of $\infty$-function theory, the assertion that the tensor product of $\infty$-functions is already “completed”

$C(X) \otimes C(Y) \simeq C(X \times Y)$

for $X$, $Y$ our generalized spaces and $C(X)$ the generalized functions on them,

is of course structurally the statement satisfied by Moerdijk-Reyes’ [[generalized smooth algebras]] which we enjoyed talking about here a while back.

There are more similarities: every generalized smooth algebra is a directed colimit of finitely generated and finitely presented ones, hence an [[ind-object]] of finite ones.

There should be a more direct relation between the two.

In particular, these generalized smooth algebras fit immediately into the concept of [[structured generalized space]]s..

What’s the bigger story that wants to be told here?

(By the way, has anyone seen Todd, lately…?)

Posted by: Urs Schreiber on May 12, 2009 6:31 PM | Permalink | Reply to this

### Re: Journal Club – Geometric Infinity-Function Theory – Week 3

Ahm, I mean apart from that I should look again in David Spivak’s thesis … :-)

Posted by: Urs Schreiber on May 12, 2009 6:41 PM | Permalink | Reply to this

### Re: Journal Club – Geometric Infinity-Function Theory – Week 3

Urs wrote:

(By the way, has anyone seen Todd, lately…?)

Todd called me a few days ago because his son had an assignment in school which required him to ask a mathematician about what math is good for in daily life.

So, he is alive and well.

Posted by: John Baez on May 12, 2009 9:57 PM | Permalink | Reply to this

### Re: Journal Club – Geometric Infinity-Function Theory – Week 3

Todd is also looking forward to the day in the not-very-distant future when he will not be so swamped with school-related activities, and have time to participate here more. (‘School’ here means elementary school.)

He may have time to make a teeny comment about something in Bruce’s wonderful post though.

Posted by: Todd Trimble on May 12, 2009 10:54 PM | Permalink | Reply to this

### Journal club at the nLab

Hi everyone,

I would like to point out a new, exciting feature at the nLab!

This is not a response to Todd’s post, although I also look forward to the not-very-distant future when you are back in full blogging mode. I really just wanted this to show up at the bottom of the page because for people who scan the NCC (that’s N-category cafe) in the old-fashion way that I do (i.e. not as a feed of some sort)

Ok, so the first line was just bait. There is no new “feature” and the “exciting” part was just to get you to keep reading. But just so you do not think I am a complete liar, there is something “new”.

I have started to use my page Alex Hoffnung as a way to keep myself on track while attempting to do way too many things. This way I can pick up a paper, read the first three pages, get discouraged, put it down, and here is the new part…come back to it someday!

So the plan is that whenever I start reading something interesting or trying to learn something new then I will do it in semi-public on my page. This goes for aspects of the JOURNAL CLUB as well as anything else I feel like typing. As things become more concrete in my head, I will make them more concrete on the nLab by exporting them to their very own pages.

I am pointing this out mostly as an invitation for anyone who feels like scribbling on my page to feel free to do so. If you want to add to or correct anything I have written, then do so without inhibition. Just try to introduce yourself by saying something like, “This is Anita Knapp…” followed by your comment. This page is completely informal and there is NO need for precision. The point is just to have a never ending piece of scratch paper with a really good eraser.

Posted by: Alex Hoffnung on May 13, 2009 5:13 PM | Permalink | Reply to this

### Re: Journal club at the nLab

Ok, now for a truly off-topic post. Last one, I promise.

Urs has just informed me how unfunny my already unfunny “I need a nap”/ “Anita Knapp” joke is to people who do not pronounce words with an American accent.

I should definitely stop being not funny. :)

Posted by: Alex Hoffnung on May 13, 2009 5:37 PM | Permalink | Reply to this

### Re: Journal club at the nLab

Urs has just informed me how unfunny my already unfunny “I need a nap”/ “Anita Knapp” joke is to people who do not pronounce words with an American accent.

Well, I just didn’t get it!

But after you explained it I had a bit of fun trying out how you guys are pronouncing this standard German name.

Posted by: Urs Schreiber on May 13, 2009 5:43 PM | Permalink | Reply to this

### Re: Journal club at the nLab

Urs wrote:

But after you explained it I had a bit of fun trying out how you guys are pronouncing this standard German name.

English-speaking mouths are not sufficiently flexible to pronounce a word starting with ‘kn’. Once we get going we can pronounce this combination of consonants: for example, nobody has trouble saying ‘acne’. But if you put it at the very start of a word, as in ‘Knapp’, we think this must be some sort of a joke: an obviously unsayable name, ha-ha, what will those Germans try next? So, we skip the ‘K’ and just say ‘napp’.

Posted by: John Baez on May 14, 2009 8:56 PM | Permalink | Reply to this

### Re: Journal club at the nLab

So, we skip the ‘K’ and just say ‘napp’.

Yes, I figured that. As we say:

Knapp daneben ist auch vorbei.

Posted by: Urs Schreiber on May 14, 2009 9:27 PM | Permalink | Reply to this

### Re: Journal Club – Geometric Infinity-Function Theory – Week 3

Before I begin, I hope everyone saw Alex’s announcement, as he wanted to keep it at the bottom of the comment page for a little while where everyone can see it.

This is a comment in response to a bit of point-set topology in Bruce’s post that caught my eye. I guess it really has to do with the terminology ‘compact’. I once asked Peter Freyd why the word ‘compact’, and IIRC he told me the word is supposed to hark back to the category of representations of a compact group, which is a compact monoidal category.

Bruce wrote

There’s a nice example: a topological space $X \in Top$ is compact if and only if the hom-functor $Hom(X,\cdot): Top \to Top$

commutes with all filtered colimits. I’d like to understand this example!

That certainly resonates with me: this seems connected with some point-set results which deserve to be in a point-set topology Hall of Fame. Literally interpreted, I don’t think the statement is quite correct, though, and I would really like to find out what the result should be, because I think something like this must be correct (as well as very appealing and suggestive).

I had to do some digging here: there is a counterexample given on page 49 in Hovey’s Model Categories, which turns out to be itself mistaken and listed among the errata to his book. The corrected counterexample (due to Don Stanley) says that the two-element set with the indiscrete topology is a compact space $X$ for which

$hom(X, -): Top \to Top$

doesn’t preserve filtered colimits, in fact not even colimits of sequences (functors out of the ordered set of natural numbers).

For example, consider the sequence of spaces

$X_n = [n, \infty) \times \{0, 1\}$

where the open sets are of the form

$[n, \infty) \times \{0\} \cup [m, \infty) \times \{1\}$

(where $m \geq n$), plus the empty set. Define $X_n \to X_{n+1}$ so that it sends a pair $(k, \varepsilon)$ to itself if $k$ > $n$, and $(n, \varepsilon)$ to $(n+1, \varepsilon)$. This defines a functor

$F: \mathbb{N} \to Top$

The colimit $X_{\infty}$ of this sequence is the two-element set $\{0, 1\}$ with the indiscrete topology. However, the identity map on this space does not factor through any of the canonical maps $X_n \to X_{\infty}$. It follows that the comparison map

$colim_n hom(X_{\infty}, X_n) \to hom(X_{\infty}, X_{\infty})$

is not surjective, and therefore not an isomorphism.

I don’t know if the story is any different for $X$ compact Hausdorff, but it could be worth considering.

One of the “hall-of-fame” results I had in mind is given on page 50 of Hovey’s book: if $Y$ is compact, then $hom(Y, -)$ preserves colimits of functors mapping out of limit ordinals, provided that the arrows of the cocone diagram,

$X_\alpha \to X_\beta,$

are closed inclusions of $T_1$ spaces. (This applies for example to the sequence of inclusions of n-skeleta in a CW-complex. Taking $Y = S^k$, this has obvious desirable consequences for the functor $\pi_k$.) Hovey wants this result in view of a small object argument on the way to proving that $Top$ is a model category. I’ll bet one can beef it up to a similar result but for more general filtered diagrams.

I call it a hall-of-fame result because (a) it’s quite useful; (b) it’s not entirely trivial to prove. Further precise results along these lines would be most welcome.

Posted by: Todd Trimble on May 13, 2009 10:35 PM | Permalink | Reply to this

### Re: Journal Club – Geometric Infinity-Function Theory – Week 3

Perhaps $Top(X,-): Top \longrightarrow Set$ was meant in the above?

Then this functor can only preserve filtered colimits if the space $X$ is actually discrete.

Let $W$ be any subset of $X$ and consider $Y_i = X \cup N$ (disjoint union) for $i \in N$ where the only nontrivial open sets are those of the form $W \cup \{n | n\geq k \}$ for $k\geq i$. with $Y_i \longrightarrow Y_{i+1}$ the identity.

Then $Y=colim Y_i$ is indiscrete. If the obvious inclusion of $X$ into $Y$ factors through some $f: X \longrightarrow Y_i$, then $W = f^{-1}(W \cup \{n | n\geq i\})$ is open.

Posted by: Marc Olschok on May 17, 2009 12:15 AM | Permalink | Reply to this

### Re: Journal Club – Geometric Infinity-Function Theory – Week 3

Perhaps $Top(X, −): Top \to Set$ was meant in the above?

It’s true that I was really only using the ordinary (set-valued) hom-functor in my comment above, although we can replace $Set$ by $Top$ if we assume $X$ is exponentiable (meaning $X \times -: Top \to Top$ has a right adjoint), or if $Top$ is interpreted liberally as denoting some ‘nice’ category of spaces assumed to be cartesian closed. I’m wondering whether Bruce had such a nice variant of $Top$ in mind.

The construction you give looks clever, but I’m not seeing why the colimit is codiscrete (I mean indiscrete). The underlying set of the colimit is $latex X \cup \mathbb{N}$, and it looks to me like $W$ is open in the colimit, since its inverse image with respect to all of the maps

$Y_i \to colim_i Y_i$

is open in $Y_i$ by definition.

I still feel that something along the lines of Bruce said could well be correct by tweaking the hypotheses a bit, maybe by working with the “right” category of spaces, possible compactly generated Hausdorff spaces for instance. Somehow the idea of an internal covariant hom-functor preserving filtered colimits is tied in my mind with a finiteness property (cf. the situation for $Vect$), and compactness is intuitively a topological finiteness property. I’d like to can come back to this and make it more precise.

Posted by: Todd Trimble on May 17, 2009 2:31 AM | Permalink | Reply to this

### Re: Journal Club – Geometric Infinity-Function Theory – Week 3

I haven’t really been following this discussion, but one thing you can say is that a space $X$ is compact iff $hom(X,-):O(X) \to Set$ preserves filtered colimits, where $O(X)$ is the poset of open subset of $X$. I think it follows that if $hom(X,-)$ preserves filtered colimits in any category of spaces that contains $O(X)$, then $X$ is compact.

But in general, defining compactness/finiteness by homming into filtered colimits works best in locally presentable / accessible sorts of categories, which $Top$ is not, nor is the category of compactly generated spaces. One might be able to characterize the sequentially compact subsequential spaces in this way, though.

Posted by: Mike Shulman on May 17, 2009 7:03 AM | Permalink | Reply to this

### Re: Journal Club – Geometric Infinity-Function Theory – Week 3

Quite right, Mike. (Which reminds me: the nLab entry on locally presentable categories could probably stand some beefing up. I really don’t know that stuff as well as I’d like.)

Any feeling on whether the iff result for subsequential spaces is actually true?

Posted by: Todd Trimble on May 17, 2009 6:03 PM | Permalink | Reply to this

### Re: Journal Club – Geometric Infinity-Function Theory – Week 3

Actually, I think it’s still false for subsequential spaces unless you at least restrict to colimits of subspace inclusions. The same sorts of counterexamples tend to apply.

It’s easy to show that a sequentially compact subsequential space is compact relative to colimits of sequences of subspace inclusions (Hovey’s argument applies without the need for $T_1$ or closedness). But I don’t know about more generally or whether there is a converse.

Posted by: Mike Shulman on May 18, 2009 4:31 AM | Permalink | Reply to this

### Re: Journal Club – Geometric Infinity-Function Theory – Week 3

a space $X$ is comopact iff $hom(X,-) Op(X) \to Set$ preserves filtered colimits.

Yes, that must be it. Or slightly more generally: any $U \subset X$ is a compact topological space precisely if $Hom_{Op(X)}(U,-)$ preserves filtered colimits.

Compare example 5.3.4.3 of Lurie’s book.

I have tried to clean up the entry [[compact object]] accordingly.

Posted by: Urs Schreiber on May 17, 2009 7:23 PM | Permalink | Reply to this

### Re: Journal Club – Geometric Infinity-Function Theory – Week 3

The only nontrivial open subsets of $Y_i$ are the sets

$D_i(k) = W \cup \{n | n\geq k \}$

for $k\geq i$.

Such a $D_i(k)$ looks like $W$ placed beside a principal upper set of $\mathbb{N}$, but you cannot have $W$ alone. More important, this setup guarantees that $D_i(k)$ will not be open in $Y_{k+1}$ or at any later stages.

In particular, no nontrivial subset of the colimit can be open in all the $Y_i$.

Posted by: Marc Olschok on May 17, 2009 11:10 PM | Permalink | Reply to this

### Re: Journal Club – Geometric Infinity-Function Theory – Week 3

Oh! Sorry, I misread what you wrote.

That’s very interesting, and the basic idea seems to me flexible enough that many a conjecture along the lines of what was being floated before could probably easily be squashed. Thanks!

Posted by: Todd Trimble on May 17, 2009 11:48 PM | Permalink | Reply to this

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