Question on Synthetic Differential Forms

August 3, 2009

Question on Synthetic Differential Forms

Posted by Urs Schreiber

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I am thinking about cosimplicial objects in the category of generalized smooth algebras as models for the $(\infty,1)$ -quantities (see there for what I mean) dual to models for $\infty$ -stacks.

The archetypical example should be the cosimplicial generalized smooth algebra of differential forms on a smooth space

for $X$ a smooth space, let $C^\infty(X^{\Delta_{inf}^k})$ be the generalized smooth algebra of functions on the infinitesimal $k$ -simplices in $X$ . As $k$ varies these naturally form a cosimplicial object.

Now, it seems to me as if the standard definition of the cochain complex of differential forms in synthetic differential geometry is nothing but the image of this under the (dual) Dold-Kan correspondence: $\Omega^\bullet(X)$ is the normalized (dual) Moore complex as recalled in this dual form and in the present context in section 4 of

Castiglioni-Cortiñas: Cosimplicial versus DG-rings: a version of the Dold-Kan correspondence.

Details of what I have in mind are at differential forms in synthetic differential geometry.

My question is: am I hallucinating? If not, has this been discussed elsewhere`?

Posted at August 3, 2009 12:44 PM UTC

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Chern 1944; Re: Question on Synthetic Differential Forms

Didn’t Chern replace a laborious proof using simplicial complexes with his amazing new approach linking curvature invariants to characteristic classes in S.-s. Chern, A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Ann. of Math. 45 (1944), 747-752. MR 0011027 (6:106a)? Isn’t that what needs to be n-Catgorified?

Posted by: Jonathan Vos Post on August 3, 2009 9:18 PM | Permalink | Reply to this

What is this about?

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Isn’t that what needs to be $n$ -Catgorified?

What do you mean?

My question was:

has anyone considered the fact (if indeed it is a fact and not a hallucination) that

the deRham complex of differential forms on $X$

the dual normalized Moore complex

the cosimplicial smooth algebra

functions on infinitesimal simplices in $X$ ?

Posted by: Urs Schreiber on August 4, 2009 8:42 AM | Permalink | Reply to this

Re: Chern 1944; Re: Question on Synthetic Differential Forms

I’m afraid, Jonathan, that your remark makes no sense. Urs is surely familiar with Chern’s “amazing new” ideas from 1944, but that’s not what’s he’s talking about.

Posted by: John Baez on August 4, 2009 10:32 AM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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I certainly have never thought or heard about this idea of yours, Urs, but it sounds believable to me.

There are some portions of this idea that are new to me, but make me feel I should have already known them.

For example, I know and love the Dold–Kan theorem in this guise:

Theorem 1: the category of simplicial abelian groups is isomorphic to the category of nonnegatively graded chain complexes of abelian groups.

In fact there’s nothing terribly special about abelian groups here; the proof just requires some elementary stuff about kernels, direct sums, and so on. So it at least generalizes this far:

Theorem 2: given an abelian category $A$ , the category of simplicial objects in $A$ is isomorphic to the category of nonnegatively graded chain complexes in $A$ .

(I leave it to Zoran Skoda to say what happens when $A$ is semiabelian: people definitely do think about simplicial groups that are not necessarily abelian.)

Anyway, for some reason I’d never thought about the ‘dual’ Dold–Kan theorem stated in the first sentence of Castiglioni and Cortinas’ paper:

Theorem 3: the category of cosimplicial abelian groups is isomorphic to the category of nonnegatively graded cochain complexes of abelian groups.

Which is silly of me, because it follows instantly from Theorem 2 by taking $A = AbGp^{op}$ . Indeed, just as instantly, Theorem 2 implies

Theorem 4: given an abelian category $A$ , the category of cosimplicial objects in $A$ is isomorphic to the category of nonnegatively graded cochain complexes in $A$ .

Anyway, I think I almost understand how Theorem 2 paves the way towards the result in the second sentence of Castiglioni and Cortinas’ paper:

Theorem 5: the model category of cosimplicial rings is Quillen equivalent to the model category of differential graded rings (nonnegatively graded, with differential of grade $1$ ).

I bet the point is that both [cosimplicial abelian groups] and [nonnegatively graded cochain complexes of abelian groups] have an ‘obvious tensor product’ making them into a monoidal category, and then:

Theorem 6: the category of monoids in [cosimplicial abelian groups] is equivalent to the category of cosimplicial rings.

and

Theorem 7: the category of monoids in [nonnegatively graded cochain complexes of abelian groups] is equivalent to the category of differential graded rings (nonnegatively graded, with differential of grade $1$ ).

Then Theorem 5 would be trivial if [cosimplicial abelian groups] and [nonnegatively graded cochain complexes of abelian groups] were equivalent as monoidal categories.

And this in turn would be trivial if [simplicial abelian groups] and [nonnegatively graded chain complexes of abelian groups] were equivalent as monoidal categories — taking the opposite category can’t hurt here.

But it’s a famous fact that [simplicial abelian groups] and [nonnegatively graded chain complexes of abelian groups] are not quite equivalent as monoidal categories, if we give each one its ‘obvious’ tensor product.

Instead, the equivalence of these two categories only preserves the tensor product ‘up to homotopy’. This is the fact lurking behind the Eilenberg–Zilber theorem: note the phrase ‘ $G F$ is chain-homotopic to the identity’ here.

So, [cosimplicial abelian groups] and [nonnegatively graded cochain complexes of abelian groups] are not equivalent as monoidal categories, but they are so ‘up to homotopy’. And this is what Castiglioni and Cortinas mention “equivalences of monoidal model categories” in the fourth sentence of their paper.

So, I get the basic idea of Theorem 5, and I think I’ll stop here, since this has taken way too long to write. I’ll try to convince myself that it wasn’t a waste of time by putting this comment on the $n$ Lab.

“No, Mom, I’m not wasting my time blogging! I’m contributing to the $n$ Lab!”

Personally I would also enjoy seeing that [simplicial abelian groups] and [nonnegatively graded chain complexes of abelian groups] are equivalent as monoidal 2-categories, if this is in fact true. Or as $(\infty,1)$ -categories.

Posted by: John Baez on August 4, 2009 11:26 AM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

Well, now I see that most of this stuff is already on the nLab, including a reference to a result by Dominique Bourne on the semiabelian case… a grand generalization of the Cegarra–Carrasco theorem, I guess!

That nLab page is quite thorough… you’ve really been thinking a lot about Dold–Kan, Urs!

But it was still fun thinking about it myself.

Posted by: John Baez on August 4, 2009 11:40 AM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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Thanks for your replies!

Well, now I see that most of this stuff is already on the $n$ Lab,

I tried to make a post where each and every technical term was linked to its $n$ Lab page. My idea is that this should enlarge the potential radius of readers, as every reader can just click on any keyword about which he or she is wondering (either way: because its unfamiliar or because it sounds as if I am spouting nonsense).

But probably the result is that there are too many links in total and none of them is actually recognized and followed.

Which one is it? Would it help if I didn’t make entire words a hyperlink but followed each index word with a hyperlinked arrow as in

keyword ( $\to$ )

as I do sometimes?

That $n$ Lab page is quite thorough… you’ve really been thinking a lot about Dold–Kan, Urs!

I spent some energy on that page, yes, but this page has seen a lot of interaction and different contributors. It was fun. Eventually it needs to be merged better with the entry on the Moore complex ( $\to$ ). Currently the two entries can’t quite decide who is in charge of what precisely.

I certainly have never thought or heard about this idea of yours, Urs, but it sounds believable to me.

Okay, good. In view of the above I should say again that all the details of the argument are given at differential forms in synthetic differential geometry (#).

And its a simple argument: I just go through Anders Kock’s characterization of synthetic differntial forms and say

a) hey, this condition that the functions have to vanish on degenerate simplices just say that they are in the joint kernel of the degeneracy maps of the cosimplicial object $C^\infty(X^{\Delta^\bullet_{inf}})$ .

b) that the coboundary operator is just the alternating sum of the face maps of this cosimplicial object

c) that comparing this with the definition of the dual normalized Moore complex in the dual Dold-Kan correspondence (see there) manifestly shows that $\Omega^\bullet(X)$ is just this: the dual normalized Moore complex of that cosimplicial object.

So it actually looks like a pretty obvious and easy to check statement. I am just looking for a sanity check.

Personally I would also enjoy seeing that [simplicial abelian groups] and [nonnegatively graded chain complexes of abelian groups] are equivalent as monoidal 2-categories, if this is in fact true. Or as (∞,1)-categories.

Yes, it should actually induce an equivalence of $(\infty,1)$ -categories. With the standard model category structure on simplicial abelian groups and chain complexes one part of the classical theorem says that both the nerve and the Moore complex functor preserve weak equivalences. So for a Quillen equivalence it would be sufficient to check that the nerve preserves fibrations. I guess that’s obvious one you chose the projective model structure on chain complexes.

Posted by: Urs Schreiber on August 4, 2009 12:31 PM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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Urs wrote:

I tried to make a post where each and every technical term was linked to its nLab page. My idea is that this should enlarge the potential radius of readers…

But probably the result is that there are too many links in total and none of them is actually recognized and followed.

Which one is it?

I’m sure different people have different opinions so I’ll just state mine: treat it as a single data point.

Your original post was so full of hyperlinked buzzwords that I decided to ignore it. I thought something like “Wow, Urs is sure becoming intimidatingly familiar with simplicial techniques, synthetic differential geometry and so on. I wonder if anyone will understand what he’s saying and post a reply?”

Then Jonathan Post posted a reply that had nothing to do with your question.

That had a good side-effect: it caused you to restate your question in a cute style consisting mainly of 4 long phrases. Sort of like you were talking slowly and loudly to someone who was old and rather deaf.

This caught my attention: I thought something like “Well, I bet Jonathan won’t understand this stuff, but maybe I can.”

And then I realized I could!

I decided to see if what you were saying seemed true. I didn’t feel like reading about the Dold–Kan theorem (since I felt I understood that theorem), so I didn’t click on that link (though it would have been helpful). Instead I was puzzled about what the dual Dold-Kan correspondence might say, and how it would relate cosimplicial rings to DGA’s. So, I peeked at the first few sentences of the Castiglioni-Cortiñas paper.

And then I started thinking, and realized that the usual Dold-Kan theorem was also a theorem about cosimplicial objects in an abelian category — something I never realized.

And then I figured, “Oh, so then monoids in cosimplicial objects in $AbGp$ should be the same as monoids in cochain complexes in $AbGp$ — that’s what going on here!”

But internal monoids need a monoidal category to live in. And then I remembered that Dold–Kan doesn’t give an equivalence of monoidal categories, just an equivalence ‘up to homotopy’ — roughly speaking.

And then I saw that reference to “equivalences of monoidal model categories” and decided I understood what you were saying and roughly why it was true (since the rest already made sense).

In short: I read your post and the first few sentences of Castiglioni-Cortiñas, and had quite a fun time figuring out what you were saying from those clues. I don’t think I wanted more information than that!

Posted by: John Baez on August 4, 2009 2:46 PM | Permalink | Reply to this

wedge product from shuffle product

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So far I had given an argument for why the deRham complex $\Omega^\bullet(X)$ is the dual normalized Moore complex of the cosimplicial algebra $C^\infty(X^{\Delta^\bullet_{inf}})$ as a complex.

I hadn’t actually checked that they agree as monoids under the lax monoidalness of the dual Dold-Kan correspondence.

So recall how Anders Kock explains that with (the joint kernel of the degeneracy maps) of $C^\infty(X^{\Delta^k_{inf}})$ identified with synthetic $k$ -forms, the wedge product on differential forms has the super-simple form as displayed for instance on page 119 of his latest book.

In particular the statement is that this simple formula indeed yields again a synthetic form, so in particular this is automatically skew symmetric in all arguments (that’s Kock’s simple but magic and powerful insight into synthetic differential forms: defined as functions on infinitesimal simplices (in the joint kernel of the degeneracy maps) both linearity as well as skew-symmetry are consequence not axioms – that’s what makes the entire discussion in this thread here possible in the first place).

What could this operation possibly correspond to in the monoidal Dold-Kan correspondence? It must be the shuffle product that gives the components of the lax compositor (=multiplicator) of the the Moore complex.

Unfortunately in Castiglioni and Cortinas’ paper this is not recalled explicitly, they just point to the formula in Schwede and Shipley, where it is on page 8, right the first point of section 2.3.

I am at a train station without pen and paper and time available, but this sure looks like it gives precisely Kock’s formula when applied to the cosimplicial algebra $C^\infty(X^{\Delta^\bullet_{inf}})$ . Just notice that by his theorem in this case all summands in the sum over unshuffles are equal. Then notice that the trivial unshuffle yields precisely his formula – unless my am-on-the-jump-combinatorics is off.

Posted by: Urs Schreiber on August 5, 2009 4:39 PM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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…my am-on-the-jump-combinatorics…

Had to run after this last statement and would have missed my train – were it not for the fact that one can rely on the German railway in that it is late with high probability.

So I have a few more minutes:

what I really wanted to say is: I would like to see whether or not it is true that

- for $A$ a smooth $\infty$ -groupoid modeled by a simplicial concrete sheaf (on some category of smooth test spaces)

- the cosimplicial copresheav

$C^\infty_{loc}(A)$

obtained by taking degreewise smooth algebras of stalks of functions at the identity cells (i.e. at the totally degenerate simplicies).

whether the correspondind normalized Moore complex differential graded algebra is always weakly equivalent to a quasi-free differential graded algebra, i.e. to the Chevalley-Eilenberg algebra of an $L_\infty$ -algebroid – as one would expect.

Must be true. I guess using the above comment on the shuffpe/wedge product one can show this…

All right, have to run once again…

Posted by: Urs Schreiber on August 5, 2009 5:06 PM | Permalink | Reply to this

synthetic Chevalley-Eilenberg

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There is a variant of this characterization of differential forms which gives a synthetic version of the Chevalley-Eilenberg algebra of the Lie algebra $g$ of a Lie group $G$ .

The two crucial input observations are the following:

a) for $G$ a Lie group let $\mathbf{B}G$ be the realization of its delooping as a smooth groupoid given by a simplicial presheaf: in degree $k$ evaluated on a smooth test space $U$ it is $\mathbf{B}G(U) = Hom_{Diff}(U,G^{\times_k})$ . Then the (non-normalized) dual Moore complex of the cosimplicial smooth function algebra

$C^\infty(\mathbf{B} G)$

is the cochain complex that computes smooth Lie group cohomology.

b) when restricted to neighbourhoods of the identity, Lie group cohomology coincides with the corresponding Lie algebra cohomology.

Classical fact, for instance: S. Świerczkowski, Cohomology of group germs and Lie algebras Pacific J. Math 39(2) 1971.

So choose a tiny open neighbourhood $V \subset G$ of the identity in $G$ . Let $\mathbf{B}G|_V$ be objectwise the sub-simplicial set on edges that sit in $V$ . This is no longer a Kan-complex valued simplicial set, but its Kan-fibrant replacement reproduces $\mathbf{B}G$ iff $G$ was simply connected. So it is just as good a model for the $\infty$ -stack in question as $\mathbf{B}G$ is.

Now, the cosimplicial smooth algebra $C^\infty(\mathbf{B}G|_{V})$ is the one whose Moore cochain complex computes the local Lie group cohomology.

Let $CE(g)$ be the Chevalley-Eilenberg algebra of $g = Lie(g)$ . regard this as a cosimplicial smooth algebra under dual Dold-Kan. There is a standard morphism

$C^\infty(\mathbf{B}G|_V) \to CE(g)$

obtained by differentiating each function at the degenerate cells of $\mathbf{B}G|_V$ , i.e. by evaluating it on synthetic infinitesimal cells of $\mathbf{B}G|_V$ .

By the above classical result it should be true that as we take the colimit over smaller and smaller $V$ this morphism becomes an isomorphism in cohomology.

Do we know if there is some tiny fixed $V$ such that we can skip the colimit and get the result nevertheless? I’d imagine that smoothly contractible $V$ should be sufficient. Is it?

I would like to say that there is a simplicial presheaf $\mathbf{B}G|_V$ weakly equivalent to $\mathbf{B}G$ such that the cosimplicial smooth algebra $C^\infty(\mathbf{B}G|_V)$ is weakly equivalent to the Chevalley-Eilenberg algebra of $g$ .

Posted by: Urs Schreiber on August 4, 2009 2:01 PM | Permalink | Reply to this

Re: synthetic Chevalley-Eilenberg

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Urs wrote:

By the above classical result it should be true that as we take the colimit over smaller and smaller $V$ this morphism becomes an isomorphism in cohomology.

Do we know if there is some tiny fixed $V$ such that we can skip the colimit and get the result nevertheless? I’d imagine that smoothly contractible $V$ should be sufficient. Is it?

I don’t have time to properly understand what you’re saying right here — but intuitively speaking, it feels right.

I’m actually imagining something like this: try to show that the Lie algebra of $G$ , viewed as a ‘simplicial infinitesimal smooth space’ sitting at the identity element of $G$ , is ‘smoothly homotopy equivalent’ to $B G|_V$ when $V$ is a small contractible neighborhood of the identity. And then somehow use this to show what you want.

I’m not sure exactly what I mean by any of this, but I have mental image of it and I like it.

On a different note: have you ever read this paper by Brylinski? It has a concept of ‘local cohomology’ which reminds me of what you’re doing. Take a look at the ‘more abstract construction’ at the bottom of page 9. It may or may not be relevant.

Sorry, these are very sketchy remarks…

Posted by: John Baez on August 4, 2009 3:19 PM | Permalink | Reply to this

Re: synthetic Chevalley-Eilenberg

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John suggested:

Take a look at the “more abstract construction” at the bottom of page 9. It may or may not be relevant.

Yes, but there he models an infinitesimal space by its topos of sheaves.

Currently I don’t feel like this petit topos approach to generalized spaces (one topos per generalized space) is the way to go. I am into the gros topos perspective (each generalized space an object in the topos).

Maybe I am wrong about this, but right now I don’t want to follow Brylinski down that particular road.

Posted by: Urs Schreiber on August 5, 2009 10:54 PM | Permalink | Reply to this

Simplices vs Cubes (again)

Is there anything in this idea that is specific to simplices, or would the same discussion go through with cubes?

Posted by: Eric Forgy on August 4, 2009 3:18 PM | Permalink | Reply to this

Re: Simplices vs Cubes (again)

Everything always also works with cubes, but cubes…

Just kidding. You may or may not know, but there’s been a war between the cubists and the simplicialists in mathematics for a long time, and the simplicialists have been winning for a long time, basically because the category of simplices has lots of amazing and convenient properties. So, cubists always say “but you can also do it with cubes!” and the majority says “great” and goes on working with simplices, and the cubists get pissed off.

So, I was just having fun teasing you.

Posted by: John Baez on August 4, 2009 5:11 PM | Permalink | Reply to this

Re: Simplices vs Cubes (again)

I suspect (with little more than gut feeling) cubists might gain some ground when it comes to directed spaces (such as spacetime).

I was recently skimming Kock’s book and saw some discussion of various shapes, but didn’t dig deep enough to make the connection here (hence my question). Thanks!

In my experience, for what it is worth, if you are doing topology, simplices are great due to their neat combinatorial properties, but are less natural when doing geometry. For example, there is no “discrete differential geometry” on simplices. You have to go to directed cubes, a.k.a. diamonds. That fact tripped me up for several years in grad school. It was a hard fought lesson that makes me ask these questions whenever I see attention focused on simplices.

Posted by: Eric Forgy on August 4, 2009 5:55 PM | Permalink | Reply to this

Re: Simplices vs Cubes (again and again)

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Eric, have a look at Anders Kock’s book (pdf online available) of which page and verse are referenced simplicially and cubically at the $n$ Lab entry. Anders Kock gives a nice and detailed discussion of the relation of the two (actually three) models.

The punchline is: you are entitled to think of each simplex as spanning a cube and all goes through as expected.

If you have access to it, the Moerdijk-Reyes book, in section 4, discusses standard facts of differential geometry using (just) synthetic cubes.

The main point of the simplices is that they model $\infty$ -groupoids more directly than cubes do. Whereas cubes tend to be easier to work with for various concrete algbraic construtions.

Posted by: Urs Schreiber on August 4, 2009 5:22 PM | Permalink | Reply to this

Manifold doesn’t care how you subdivide it; Re: Question on Synthetic Differential Forms

I was trying to get at the way that local (curvature, smoothness) data and global topology are connected, which is what was revolutionary in 1944 with Chern, whose work I of course expected almost everyone here to know. I have been working on my own (with some minor results published in reputable online venues sch as Mathworld and OEIS) for some time with simplicial complexes including polytetrahedra and polypentatopes (for Wick rotation investigations) and polyhedral complexes. Again, one can triangulate manifolds, sometimes (not always) tetrahedralize them, carve them up into simplexes, carve them up into cubes, but the goal seems to be a moral equivalent (hence the “up to homotopy”) of finding topological invariants (Betti number, Chern classes) that don’t care about the dissection.

Thank you, John Baez et al., for making a useful response to what may have seemed (or been) a stupid question of mine. I am old – compared to my college and high school students, and in how long I’ve grappled with elementary Categories (late college, early grad school, and then again 3 decades later). I am not deaf (although sometimes slow to grasp someone else’s implications), but I have a brother with a 90 decibel loss, who made me aware of sophisticated communications sub-cultures. As a teacher, I believe that there are no stupid questions, and I’m always willing to make by ignorance obvious for the benefit of mutual communications.

Posted by: Jonathan Vos Post on August 4, 2009 7:14 PM | Permalink | Reply to this

Re: Manifold doesn’t care how you subdivide it; Re: Question on Synthetic Differential Forms

Jonathan wrote:

I was trying to get at the way that local (curvature, smoothness) data and global topology are connected, which is what was revolutionary in 1944 with Chern, whose work I of course expected almost everyone here to know.

Yes. If Urs or I seemed exasperated, it’s because he was asking a cutting-edge question of laser-like precision, and you responded with a very broad question that was at most vaguely related.

It’s as if a brain surgeon were asking “Do you think his tregeminal nerve is damaged near the rhomboid fossa?” and someone walked in and said “Hey, man! Is that dude’s head, like, cut open?”

And sadly, this was far from the first occasion where your comments have been of limited relevance to the subject being discussed. So, my frustration sort of boiled over.

As a teacher, I believe that there are no stupid questions…

This is certainly a good thing to tell students in the classroom — but if someone took this as a general rule that applied in every situation, he’d probably wind up regretting it…

… unless he supplemented the rule with some fine print: there are no stupid questions, but there may be unwise times to ask.

Posted by: John Baez on August 6, 2009 5:35 PM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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Does anyone in fact have a formula/reference for the lax monoidal transformation – if indeed there is such – for the (normalized) Moore complex in dual Dold-Kan, from cosimplicial algebras to connective differential graded algebras.

I have some formula that should work if the signs come out right, but don’t have the nerve (simplicial or not) to check that right now, since it’s very tedious.

Posted by: Urs Schreiber on August 5, 2009 11:03 PM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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Urs wrote:

What could this operation possibly correspond to in the monoidal Dold-Kan correspondence? It must be the shuffle product that gives the components of the lax compositor (=multiplator) of the the Moore complex.

and later:

Does anyone in fact have a formula/reference for the lax monoidal transformation – if indeed there is such – for the (normalized) Moore complex in dual Dold-Kan, from cosimplicial algebras to connective differential graded algebras?

I’m not sure I understand what you’re up to, so forgive me if these remarks miss the point.

It sounds like you’re wanting some formula involving shuffles that explains why the normalized Moore complex is a lax monoidal functor.

So, maybe you’re secretly just seeking a proof that the ‘normalized Moore complex’ functor

$K : [simplicial abelian groups] \to [chain complexes]$

is lax monoidal, by giving an explicit formula for the natural transformation

$g: K(A) \otimes K(B) \to K(A \otimes B)$

This formula indeed involves shuffles. It’s usually given in the proof of the Eilenberg–Zilber theorem.

This formula is equation (1.2) on page 3 of Peter May’s notes on Operads and Sheaf Cohomology, where he reviews the Eilenberg–Zilber theorem. It’s probably good to start on page 2 to make sure you understand all the notation. Prop. 1.4 on page 4 says that

$g: K(A) \otimes K(B) \to K(A \otimes B)$

makes $K$ into a lax monoidal functor.

In particular, he uses $K$ to mean the unnormalized Moore complex, but he says his formula for $g$ also works for $K_N$ , the normalized Moore complex.

As you’ll see, his formula for $g$ involves shuffles! He call it the ‘shuffle, or Eilenberg–Mac Lane’ map.

But maybe you know all this stuff and you’re asking for something else.

Posted by: John Baez on August 6, 2009 8:45 AM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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Yes, that’s precisely what I am looking for, but now for the co-version

$K : [cosimplicial abelian groups] \to [connective cochain complexes] \,.$

Maybe I am mixed up, but this doesn’t seem to follow as directly from the simplicial version as one might think. Does it?

Posted by: Urs Schreiber on August 6, 2009 11:26 AM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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Urs wrote:

Does it?

I thought it almost did, but I could be mixed up.

In my original reply, I said:

Take any abelian category $A$ . We always have an isomorphism of categories

$K:$ [simplicial objects in $A$ ] $\to$ [chain complexes in $A$ ]

If we take $A = AbGp^{op}$ this gives

$K:$ [cosimplicial abelian groups] $\to$ [cochain complexes]

which is what you want. (Actually you say ‘connective’ cochain complexes, but I’m always assuming here that my chain and cochain complexes are graded by nonnegative integers, so I don’t need to say that.)

So the question is: can we take the usual formula that proves $K$ is lax monoidal when $A = AbGp$ , and get it to work for other $A$ , in particular $A = AbGp^{op}$ ?

(A digression: I’m going along with you and saying $K$ is is lax monoidal, but in fact it’s strong monoidal, meaning that the ‘laxator’, or ‘laxative’, or whatever you call it:

$g: K(A) \otimes K(B) \to K(A \otimes B)$

is invertible.)

The answer to this question must be something like: when $A$ has a tensor product that’s compatible with its abelian category structure.

After all, it’s the tensor product on $A$ that makes

[simplicial objects in $A$ ]

and

[chain complexes in $A$ ]

be monoidal categories!

So here’s the question as I see it: how compatible must the tensor product in $A$ be with its abelian category structure, for

$K:$ [simplicial objects in $A$ ] $\to$ [chain complexes in $A$ ]

to be strongly monoidal according to the usual formula?

We’ll be lucky if the answer is:

All we need is for the tensor product in $A$ to distribute over direct sums.

Because this holds in $A = AbGp$ and equally well in $A = AbGp^{op}$ .

On the other hand, the answer might be:

All we need is for tensoring with any object in $A$ to be right exact, i.e. preserve direct sums and cokernels.

This would be sad because this is true in $AbGp$ but not, I think, in $AbGp^{op}$ .

(On the other hand, tensoring with any object in $AbGp^{op}$ is left exact: it preserves direct sums and kernels!)

I really think all this stuff is so beautiful that it should be nothing but what the barbarians call ‘abstract nonsense’…

Posted by: John Baez on August 6, 2009 1:45 PM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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A digression: I’m going along with you and saying K is is lax monoidal, but in fact it’s strong monoidal, meaning that the ‘laxator’, or ‘laxative’, or whatever you call it is invertible.

Is it? I wouldn’t think so. It has a one-sided inverse when degenerate elements are divided out, but the would-be inverse from the other side is just a chain homotopy inverse.

See page 9 of Schwede-Shipley.

I am inclined to think that this is a problem with your suggestion to used abstract duality.

Because $[\Delta^{op},C^{op}]$ is not equivalent to $[\Delta,C]$ but to $[\Delta,C]^{op}$ .

That means that if there were a monoidal structure on

$C : [\Delta^{op},AbGrp^{op}] \to Ch(AbGrp^{op})$

its compositor component map would be a diagram

$C(K)\otimes C(L) \leftarrow C(K \otimes L)$

in $AbGrp$ , instead of the other way around, and this can be inverted only up to chain homotopy.

Posted by: Urs Schreiber on August 6, 2009 11:54 PM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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John wrote:

A digression: I’m going along with you and saying $K$ is is lax monoidal, but in fact it’s strong monoidal, meaning that the ‘laxator’, or ‘laxative’, or whatever you call it is invertible.

Urs wrote:

Is it? I wouldn’t think so. It has a one-sided inverse when degenerate elements are divided out, but the would-be inverse from the other side is just a chain homotopy inverse.

See page 9 of Schwede-Shipley.

Whoops — you’re right! Or Peter May’s Prop. 1.4.

(By the way, Brooke Shipley used to work at the University of Chicago, where Peter May, is based, so this is all one big happy family.)

I am inclined to think that this is a problem with your suggestion to used abstract duality.

You’re right — that’s why I brought up this point: it was a key step in my argument.

So, it seems that ‘my’ isomorphism of categories

$K : [cosimplicial abelian groups] \to [cochain complexes]$

is oplax, but not lax.

So, I can use this $K$ to turn a cosimplicial coalgebra into a differential graded coalgebra.

But I cannot use it to turn a cosimplicial algebra into a differential graded algebra.

So, it seems you really do need a different functor

$K' : [cosimplicial abelian groups] \to [cochain complexes]$

And, it sounds like you have one. That’s remarkable. Do you think it’s also an isomorphism of categories, or at least an equivalence?

Posted by: John Baez on August 7, 2009 9:00 AM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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Okay, good that we clarified this, then.

There is a more pedestrian reason to see why things can’t be as easy:

for purely the reason that cochain degrees need to be preserved, it it is clear that where the shuffle map compositor for the simplicial Moore complex map uses degeneracy maps, the shuffle map compositor for the cosimplicial Moore complex map will have to use face maps.

So this means they will have to be similar but different, to the extent that $\Delta$ is similar to but different from $\Delta^{op}$ .

So, it seems you really do need a different functor

$K′:[cosimplicialabeliangroups]\to[cochaincomplexes]$

No I want the cosimplicial Moore complex functor! Nothing else.

I want to see to which extent it is monoidal.

The motivation being: I see that it is monoidal on one particular object, namely the cosimplicial algebra of functions on infinitesimal simplices.

And, it sounds like you have one.

Based on this special case example I have a proposal for a candidate compositor map that would make the cosimplicial (normalized) Moore complex functor a lax monoidal functor.

If I had spent the time chatting on the blog on doing the remaing sign checks on my formula, I would by now maybe even know for sure! :-)

Judging from all the existing work on monoidalness of versions of Dold-Kan it is clear that this functor is at least monoidal in some possibly weakened way (such as: up to chain homotopy).

It would be good to know exactly how.

First I thought that’s what Castiglioni and Cortinas do, but then I realized that they do something else.

Posted by: Urs Schreiber on August 7, 2009 9:18 AM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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Urs wrote:

There is a more pedestrian reason to see why things can’t be as easy:

for purely the reason that cochain degrees need to be preserved, it it is clear that where the shuffle map compositor for the simplicial Moore complex map uses degeneracy maps, the shuffle map compositor for the cosimplicial Moore complex map will have to use face maps.

That means nothing to me, but I’ll take your word for it… there’s probably no point in you trying to explain it.

No I want the cosimplicial Moore complex functor!

Okay, same functor

$K:[cosimplicial abelian groups] \to [cochain complexes]$

I want to see to which extent it is monoidal.

More precisely, you’re trying make it lax monoidal instead of what I did, which was to make it oplax monoidal.

Judging from all the existing work on monoidalness of versions of Dold-Kan it is clear that this functor is at least monoidal in some possibly weakened way (such as: up to chain homotopy).

Right: while the map constructed by my ponderings

$g: K(A \otimes B) \to K(A) \otimes K(B)$

is not an isomorphism, it’s still a quasiisomorphism. So $K$ probably sends cosimplicial algebras to $A_\infty$ algebras. But you should go check your signs and find a map

$h : K(A) \otimes K(B) \to K(A \otimes B)$

that makes $K$ lax monoidal, so it sends cosimplicial algebras to differental graded algebras ‘on the nose’. That would be very nice.

Posted by: John Baez on August 7, 2009 11:14 AM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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I wrote:

it it is clear that where the shuffle map compositor for the simplicial Moore complex map uses degeneracy maps, the shuffle map compositor for the cosimplicial Moore complex map will have to use face maps.

John replied:

That means nothing to me, but I’ll take your word for it… there’s probably no point in you trying to explain it.

For the record I’ll say it anyway, it’s nothing deep, just the first observation on the formulas that we keep pointing to:

the formula for the shuffle map is

$compositor : a \otimes b \mapsto \sum_{(\mu,\nu)} \pm s_n a \otimes s_\mu b \,.$

Whatever the details are: on the left we have the tensor in chain complexes, which adds degree, while on the right we have the tensor in simplicial groups, which is degreewise. So each of $s_\mu a$ and $s_\nu a$ has to separately have the degree of $a$ plus that of $b$ . That means the $s_\mu$ operations here have to be degree-raising, that is they have to be degeneracy maps, since we are talking about simplicial groups.

The same kind of argument about degrees applies to the cosimplicial case. So we need again operations that raise degree there. But now this are the face maps.

So for the cosimplicial case one expects a formula roughly of the style

$compositor : a \otimes b \mapsto \sum_{(\mu,\nu)} \pm d_\nu a \otimes d_\mu b \,,$

whatever the other details mean.

Don’t know abut you, but I find it helpful to think of this in the special case of Kock’s formula for the wedge product of synthetic forms:

you start with a function $a := \omega_p$ on $p$ -simplices and one $b := \omega_q$ on $q$ -simplices and produce a function $\omega_p \wedge \omega_q$ on $p+q$ -simplices by

- regarding $a$ as a function $d_\mu a$ on $p+q$ -simplices by taking $p$ -dimensional faces of these and then evaluating $a$ on these faces

- regarding $b$ as a function $d_\nu b$ on $p+q$ -simplices by taking $q$ -dimensional faces of these and then evaluating $b$ on these faces

then taking the plain product of functions $d_\mu a \cdot d_\nu b$ of these two functions on the space of $(q+p)$ -simplices.

More precisely, you’re trying make it lax monoidal

Yes, because all I need for eternal happpiness is that this cosimplicial Moore complex functor (which I’ll keep calling $C$ , if you allow) allows me to tranfer monoid structures $(K, \cdot)$ from cosimplicial abelian groups $K$ to monoid structures $(C(K), \wedge)$ on cochain complexes $C(K)$ by setting

$\wedge : C(K)\otimes C(K) \stackrel{compositor of C}{\to} C(K \otimes K) \stackrel{C(\cdot)}{\to} C(K) \,.$

Posted by: Urs Schreiber on August 7, 2009 12:30 PM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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Okay, I get it now.

For what it’s worth, I asked Peter May about a puzzle dual to yours, saying:

Hi!

Over at the $n$ -Café we had a question. As you know better than I, the “normalized Moore complex” functor from simplicial abelian groups to chain complexes of abelian groups can be made lax monoidal using the shuffle map. Is there also some other way to make it oplax monoidal?

and he replied:

Not that I know of. The best expert on this might be Brooke Shipley, at UIC.

Posted by: John Baez on August 7, 2009 5:28 PM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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I really think all this stuff is so beautiful that it should be nothing but what the barbarians call ‘abstract nonsense’…

Yes, but.

Unfortunately I have to interrupt working on this for the time being and concentrate on something else. I will come back to this in a few days. This is important to sort out.

Still without going into details, I just note that while all this should be just abstract nonsense, what makes it a bit more subtle and tedious to see than usual is that one needs, I think, the precise relation between $\Delta$ and $\Delta^{op}$ . Because the role of the face and degeneracy maps interchange in parts as we switch from simplicial Dold-Kan to cosimplicial. Where the simplicial shuffle product uses degeneracy maps, the cosimplicial one will need face maps (if it exsists at all). And the trouble is, I think, that this involves subtle index shifts.

There was a while ago a discussion on the cat theory mailing list on how $\Delta$ is, while different from, similar to $\Delta^{op}$ . I think when following the abstract nonsense path this has to be taken care of when deriving the formula.

So, recall that the motivation here is that I am saying that:

It looks as if Anders Kock’s formula for the wedge product on synthetic forms in their incarnations as functions on infinitesimal simplices is nothing but the monoidal structure induced via the cosimplicial shuffle product from the canonical monoidal structure on cosimplicial function algebras to that on the Moore cochain complexes.

That suggests an obvious formula for the cosimplicial shuffle product. And it seems to come out right except for a few undesired terms that however appear in pairs.

So its all a matter of checking if the signs on the pairwise appearing junk terms indeed cancel correctly.

Probably they do and everything is all right. But I don’t have the leisure to check right now.

Posted by: Urs Schreiber on August 6, 2009 8:19 PM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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I presented a strategy for proving the result you want. You’re suggesting it won’t work, but you aren’t quite saying why it won’t work. I pointed out that there’s a ‘dual normalized Moore complex’ giving an isomorphism of categories

$K : [cosimplicial abelian groups] \to [cochain complexes]$

The construction is not sneakily different from the usual normalized Moore complex

$K : [simplicial abelian groups] \to [chain complexes]$

but is in fact ‘just the same’! Namely, they’re both special cases of

$K : [simplicial objects in A] \to [chain complexes in A]$

Is there some reason you don’t like this?

There was a while ago a discussion on the cat theory mailing list on how

\Delta

is, while different from, similar to

\Delta^{op}

. I think when following the abstract nonsense path this has to be taken care of when deriving the formula.

Right — the relation between $\Delta$ and $\Delta^{op}$ is endlessly fascinating and annoying. But I’m thinking of a cosimplicial object in $A$ as a simplicial object in $A^{op}$ , and hoping this takes care of everything.

Posted by: John Baez on August 6, 2009 8:35 PM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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That was quick!

I may be mixed up. I keep finding that I need extra work to guess and check the compositor formula for the cosimplicial Dold-Kan map.

I pointed out that there’s a ‘dual normalized Moore complex’ giving an isomorphism of categories

$K : [cosimplicial abelian groups] \to [cochain complexes]$

Okay. I know that. This is what started this thread here.

I also see that at this level everything is just abstract duality.

My trouble is with the monoidal structure of $K$ . That doesn’t follow as simply.

If it does and I am mixed up, you could do me a grand favour by giving me the formula for the compositor in the cosimplicial monoidal Dold-Kan correspondence.

Yesterday I made a guess. I have now copied this from my private web to the end of this entry here:

$(\infty,1)$ -quantity

Somewhere towards the end there is an unproven proposition that starts with “There is a natural transformation…”

I want to know if that proposition is right. Or else, if it can be corrected.

Posted by: Urs Schreiber on August 6, 2009 8:48 PM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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I wonder if the combinatorics work out differently with cube categories $\Box$ and $\Box^{op}$ .

</peanut gallery>

Posted by: Eric Forgy on August 6, 2009 9:29 PM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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Urs wrote:

My trouble is with the monoidal structure of $K$ . That doesn’t follow as simply.

If you are happy with my description of the functor

$K : [cosimplicial abelian groups] \to [cochain complexes]$

that is, if you agree that we should treat this as a special case of the general construction

$K : [simplicial object in A] \to [chain complexes in A]$

then I’d be interested in what you think of my strategy for making this functor laxly monoidal.

If it does and I am mixed up, you could do me a grand favour by giving me the formula for the compositor in the cosimplicial monoidal Dold-Kan correspondence.

If I knew anything about this of course I would tell you. But since I don’t, I’m trying to make it up myself. After all, none of the really smart people are helping you with your question.

My best guess so far is that the formula you want is ‘exactly the same’ as for the simplicial case, after taking the necessary ‘op’ into account.

In particular, note that $AbGp^{op}$ is equivalent to the category of compact abelian groups, so a cosimplicial abelian group is just a simplical compact abelian group, and a cochain complex of abelian groups is just a chain complex of compact abelian groups.

So, there’s some hope that making

$K : [cosimplicial abelian groups] \to [cochain complexes]$

laxly monoidal is really shockingly similar to making

$K : [simplicial abelian groups] \to [chain complexes]$

laxly monoidal, because it just amounts to making

$K : [simplicial compact abelian groups] \to$ $[chain complexes of compact abelian groups]$

laxly monoidal

Posted by: John Baez on August 6, 2009 10:01 PM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

I’ve heard Jenny Harrison talk about a framework of differential geometry using infinitesimal chainlets (which are based on cubes, as Forgy likes), but I don’t think it is approached using any visible categorical language at all. I believe she has a couple papers on the arXiv on this subject. Beyond the usual geometry of manifolds, I think she uses it to study fractal geometries and objects of non-integer Hausdorff dimension.

Posted by: Scott Carnahan on August 10, 2009 3:33 AM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

Hi Scott,

I’m a big fan of Jenny’s stuff. We talked about it a bit here:

Harrison’s Geometric Calculus

and here:

The search for discrete differential geometry

It is still a dream of mine to see a paper co-authored by Jenny and Urs. Maybe one day…

PS: It appears Whitney’s beautiful “Geometric Integration Theory” FINALLY made it into a Dover publication.

Posted by: Eric Forgy on August 10, 2009 4:34 AM | Permalink | Reply to this

cup product

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Maybe I am hallucinating, but isn’t there a very simple answer to this question of making the Moore cochain complex functor

$C : cosimplicial abelian groups \to cochain complexes$

lax monoidal, as long as we don’t ask it to be symmetric lax monoidal?

For $K$ and $L$ two cosimplicial groups, take the compositor map

$\mu : C(K) \otimes C(L) \to C(K \otimes L)$

to be given on homogeneous elements

$a \otimes b \in C^p(K)\otimes C^q(L) \subset (C(K)\otimes C(L))^{p+q}$

$\mu_{K,L}(a \otimes b) = (d_L a) \otimes (d_R b)$

where $d_L$ is the image under $K : \Delta \to Ab$ of the initial order preserving injection $L : [p] \to [p+q]$ that is the identity on natural numbers, and where $d_R$ is the image under $L$ of the terminal order preserving injection $[q] \to [p+q]$ that as a function on natural numbers is addition with $p$ .

This way the monoid structure on the Moore cochain complex $C(K)$ of a cosimplicial ring $K$ would be that given by the product

$C(K) \otimes C(K) \stackrel{\mu_{K,K}}{\to} C(K \otimes K) \stackrel{C(-\cdot-)}{\to} C(K)$

where $-\cdot-$ denotes the product on $K$ .

This is nothing but the cup product.

It’s not graded commutative even if $K$ is commutative at this level, it becomes only graded commutative in cohomology. So $\mu$ is not symmetric on the nose, but at best in some homotopical sense. But so what?

Or maybe I am hallucinating. It’s been a long day.

Posted by: Urs Schreiber on August 20, 2009 4:50 PM | Permalink | Reply to this

Re: cup product

Sorry for a naive question, but would symmetry relate back to commutativity?

Cup product is not commutative “on the nose” at the cochain level, but that might not be relevant at all.

Oh! I should have read more than the first sentence. In the second to last paragraph you address this. One of these days you will resurrect all our old ideas and make me a happy old man :) The way it fails to be graded commutative is probably interesting…

*runs back into hiding*

Posted by: Eric Forgy on August 20, 2009 6:29 PM | Permalink | Reply to this

Re: cup product

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Sorry for a naive question, but would symmetry relate back to commutativity?

Yes, precisely. “Symmetric monoidal” is the cat theory way of saying “as commutative as possible”.

See the entry symmetric monoidal category.

(I have just expanded that a bit, but it is still very stubby, unfortunately.)

So for instance the category of cochain complexes is symmetric monoidal: you can tensor cochain complexes this way or that way, and the results are, while not equal, isomorphic.

What I tried first, the computation that I kept alluding to in the above discussion where I said that there is a tedious check on some signs, was a formula where instead of the plain cup product I’d take a “skew symmetrized” version of it. I was hoping to make the Moore cochain complex functor a morphism between symmetric monoidal categories this way.

I got kind of annoyed with the computation, though, and realized that if I just dropped the skew-symmetrization, then the formula I was trying amounted to essentialy nothing but the standard cup product.

Posted by: Urs Schreiber on August 21, 2009 1:30 PM | Permalink | Reply to this

Re: cup product

Thanks. That is interesting. Another naive thought…

This reminds of the old folklore that says a discrete differential algebra cannot be both skew commutative and associative. By skew symmetrizing, I’m guessing you lost associativity. Conversely, maintaining associativity likely costs you skew commutativity, but that is often a feature and not a bug, e.g. noncommutative geometry. But you know that because we’ve been talking about it for years :)

It is usually about this time in the conversation that I wish I was able to read and understand Jim’s stuff :)

PS: I’ve always been fond of the idea I talked about here. I wonder how it would translate into lingua categoria, e.g. “symmetric”, “monoidal”, something or other.

Posted by: Eric Forgy on August 21, 2009 3:19 PM | Permalink | Reply to this

Re: cup product

some ambiguity here:

This reminds of the old folklore that says a discrete differential algebra cannot be both skew commutative and associative. By skew symmetrizing, I’m guessing you lost associativity.

what’s `discrete’?
of course one does have graded commutative associative algebras
and
in characteristic 0, they even model topology

but that doesn’t work in char p or over Z

indeed if you graded symmetrize over Z
the singular cochain algebra
you do lose associativity - up to homotopy
so you had something else in mind?

Posted by: jim stasheff on August 22, 2009 2:20 AM | Permalink | Reply to this

Re: cup product

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what’s `discrete’?

By “discrete differetial graded algebra”, I mean a DGA generated by a finite (or possibly countable) set.

The elements in the set can be identified with points of a finite (or countable) topological space.

Although it could still use some work, I tried to explain the idea (to the best of my ability) here.

In the case of a “discrete” DGA, the degree zero subalgebra is the algebra of projections.

of course one does have graded commutative associative algebras

and

in characteristic 0, they even model topology

but that doesn’t work in char p or over Z

I’m embarrassed by a potential language barrier on my part, but I think that a graded commutative associated algebra (GCAA) can only model a continuum topology. Is that correct? Is that the same thing as what you said?

The neat thing (which is probably well known) is that graded noncommutative associated algebras (GNCAA) can model finitary topologies. Even neater, in my opinion, is that the noncommutativity disapears in the continuum limit, i.e. as you make the finitary topology dense in the continuum topology through a limiting procedure the noncommutativity disappears. This has always reminded me of the “classical limit” in quantum mechanics so that GNCAAs can be thought of as quantized versions of the GCAAs. Or, as I would rather think of it, GCAAs are “classicial limits” of GNCAAs. Then all the wonderful things that can be done with GCAAs imply wonderful thing that can be done with GNCAAs. In other words, the many wonderful things that come from GCAAs are special cases of wonderful things that come from GNCAAs.

The non-rigorous (yet) associations I think about go like this:

$\text{GCAA} \stackrel{\quad\quad\quad}{\leftrightarrow} \text{Continuum Topology}$

$\text{GNCAA} \stackrel{\quad\quad\quad}{\leftrightarrow} \text{Finitary Topology}$

$\text{GCAA}\stackrel{\text{Quantization}}{\to}\text{GNCAA}$

$\text{GNCAA}\stackrel{\text{Classical Limit}}{\to}\text{GCAA}$

$\text{Continuum Topoplogy}\stackrel{\text{Discretization}}{\to}\text{Finitary Topology}$

$\text{Finitary Topoplogy}\stackrel{\text{Continuum Limit}}{\to}\text{Continuum Topology}$

Forming the square, we have

$\begin{aligned} \text{GCAA} & \stackrel{\quad\quad\quad}{\leftrightarrow} & \quad\text{Continnum Topology} \\ \darr & {} & \darr \\ \text{Quantization} & {} & \text{Discretization} \\ \darr & {} & \darr \\ \text{GNCAA} & \stackrel{\quad\quad\quad}{\leftrightarrow} & \quad\text{Finitary Topology} \end{aligned}$

or, in what my opinion is the more natural direction

$\begin{aligned} \text{GNCAA} & \stackrel{\quad\quad\quad}{\leftrightarrow} & \quad\text{Finitary Topology} \\ \darr & {} & \darr \\ \text{Classical Limit} & {} & \text{Continuum Limit} \\ \darr & {} & \darr \\ \text{GCAA} & \stackrel{\quad\quad\quad}{\leftrightarrow} & \quad\text{Continuum Topology} \end{aligned}$

I’m hoping that Kantization can help make these ideas rigorous.

Posted by: Eric Forgy on August 23, 2009 1:59 AM | Permalink | Reply to this

Re: cup product

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I have written up the details of what I think is a proof of the

Lax monoidalness of the Moore cochain complex functor

(go to that link and scroll down and find the section with this title).

For every cosimplicial ring $K$ this induces on the Moore cochain complex $C(K)$ the structure of a differential graded algebra. The product operation is the cup product.

Check. Its an easy computation. It is almost literally the standard discussion of the cup product in singular cohomology, only that

a) it uses the fact that the standard discussion depends on nothing but the structure of a cosimplicial ring (on the functions on simplices, for instance).

b) the cup product monoidal structure is really straightforwardly refined to a lax monoidal structure on the Moore complex functor and then derives from that in the obvious manner.

This must be well known in some way (or else I am mixed up). I’d be grateful either for references that explicitly state this or for pointing out the mistake that I am making.

Posted by: Urs Schreiber on August 20, 2009 9:47 PM | Permalink | Reply to this

Re: cup product

Urs wrote:

So μ is not symmetric on the nose, but at best in some homotopical sense. But so what?

In some very specific homotopical sense! cf. Steenrod’s cup_i products
which led to his operations
and then on to …..

Posted by: jim stasheff on August 21, 2009 2:35 PM | Permalink | Reply to this

Re: cup product

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So is there some precise way in which we can think of the Steenrod square operations and refining the cup product to a homotopy coherent symmetric monoidal functor? Or something like that?

Posted by: Urs Schreiber on August 21, 2009 2:57 PM | Permalink | Reply to this

Re: cup product

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There is a relevant ancient paper of mine on Steenrod operations:

J.P. May. A general algebraic approach to Steenrod operations. Lecture Notes in Mathematics Vol. 168. Springer-Verlag 1970, 153–231.

This was in a conference proceedings for Steenrod’s 60th birthday, and he appreciated it. It shows how to get Steenrod operations in a variety of contexts; cosimplicial rings are just one special case. Another, relevant to my thesis even earlier, is Steenrod operations in the cohomology of Hopf algebras, such as the $E_2$ term of the Adams spectral sequence, which were first defined in a paper by Novikov that I read but that may never have been translated into English; they were developed by Liulevicius a little later, and I think independently.

The paper cited above should have been a paper all about operads, but in fact it was understanding of the diagrams that appear in the definition of Steenrod operations and in particular in the proofs of the Cartan formula and Adem relations that led me to the definition of operads just a little later:

The geometry of iterated loop spaces. Lecture Notes in Mathematics Vol. 271. Springer-Verlag 1972.

However, conceptually, Steenrod operations do not require the level of precision of operad actions. they are entirely a matter of up-to-homotopy versions of the operadic diagrams. I would be bored to try to cram the idea into some such jargon as “homotopy coherent symmetric monoidal functor on cochains”, but operad actions can of course be viewed in some such light.

The question of homotopy commutativity is more interesting. Mod 2, in Steenrod’s original definition, the operations come directly from the cup_i products, which directly concern higher homotopy commutativity. It was understanding of what the cup-i products were saying in terms of group cohomology that led Steenrod to the mod $p$ operations for p odd. But at odd primes to this day one does not really have a direct way of understanding the operations in terms of basic operations such as cup_i products. Operad actions encode the forest of all relevant higher homotopies, and to hell with making the individual trees of higher chain homotopies explicit.

Posted by: Peter May on August 21, 2009 6:24 PM | Permalink | Reply to this

Re: cup product

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You may recall that in the above discussion Jim Stasheff had suggested that the cup product (on, say, the dualMoore complex of functions on a simplicial set) is, while not in general commutative, in some sense homotopy-commutative and that making this homotopy commutativity explicit involves Steenrod square operations.

When I asked for a precise statement I was pointed to Peter May, whose kind reply is above.

I now find that the statement in question is discussed explicitly in

Benoit Fresse, Clemens Berger, Combinatorial operad actions on cochains

The main theorem stated on the first page there says that, indeed, there is an $E_\infty$ operad acting on the dual Moore complex of a simplicial set whose binary operation is the cup product.

And indeed, as Jim suggested, the author claims that the idea of his construction goes back to Steenrod square operations.

For whatever that’s worth, this seems to answer my question.

Posted by: Urs Schreiber on November 11, 2009 2:09 PM | Permalink | Reply to this

dualMoore complex

Urs: say, the dualMoore complex of functions on a simplicial set

what in the world is the dualMoore complex?
unless you mean the complex of cochains on a simplicial set?

Posted by: jim stasheff on November 11, 2009 2:26 PM | Permalink | Reply to this

Re: cup product

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unless you mean the complex of cochains on a simplicial set?

Yes, sure.

But I would think it is more general than that: given any cosimplicial algebra $A$ (be it functions on a simplicial set or something else) we can form its Moore cochain complex $N^\bullet(A)$ (“dual” to indicate that this is not the chain complex from Dold-Kan, but the cochain complex from “dual Dold-Kan”) and this $N^\bullet(A)$ becomes a dg-algebra using the obvious generalization of the cup product operation to arbitrary cosimplicial algebras.

Posted by: Urs Schreiber on November 11, 2009 2:36 PM | Permalink | Reply to this

Re: cup product

$MathML-enabled post (click for more details).$

I now find that the statement in question is discussed explicitly in…

It is being pointed out to me that the statement is older, going back to an article by Hinich and Schechtman from 1987.

I’ll provide a link to summary in a moment…

Posted by: Urs Schreiber on November 11, 2009 5:16 PM | Permalink | Reply to this

Re: cup product

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I have now collected some useful bits on the $\infty$ -commutativity of the cochain complex of a simplicial set at

cochains on simplicial sets

As a special treat, this features a pdf copy of a colloquium lecture by Steenrod from 1957, where the the lower degree homotopies are explicitly given by hand in terms of concrete cochains.

Many thanks to Jim Stasheff for providing these notes by Steenrod.

And many thanks to Peter May for providing an unpublished set of notes on this stuff, which is very useful indeed. I’ll try to find out if I may link to that, too.

Posted by: Urs Schreiber on November 11, 2009 8:08 PM | Permalink | Reply to this

Re: cup product

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Oh! I read the first couple of pages of Steenrod’s lectures. It’s beautiful! Its written so clearly even I can understand it. But as I read, I suddenly got excited for another reason and have to ask before I continue.

Will these lectures help me understand Kan extension?

Posted by: Eric Forgy on November 12, 2009 12:46 AM | Permalink | Reply to this

Re: cup product

I don’t know, but why don’t you just go straight to the original source?

Daniel M. Kan, Functors involving c.s.s. complexes, Trans. Amer. Math. Soc., vol. 87, no. 2 (March 1958).

Looking in retrospect more than 50 years later, the prescience of this paper is completely mind-blowing. This is where Kan extensions were originally introduced (not by the same name!). Categorical tensor products, Yoneda lemma, weighted colimits… if you look carefully, it’s basically all in there. And written by a homotopy theorist, not a category theorist per se.

This may be just for you, and I think it would amply repay careful study.

Posted by: Todd Trimble on November 12, 2009 1:33 AM | Permalink | Reply to this

Re: cup product

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By the way, direct links to this article are in the list of references here.

Posted by: Urs Schreiber on November 12, 2009 1:44 AM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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Jim Stasheff kindly points me to an article that discusses some statement along the lines discussed here:

Anette Huber, Guido Kings, A $p$ -adic analogue of the Borel regulator and the Bloch-Kato exponential map (pdf)

(You woudn’t have guessed that from the title, would you?)

The relevant bit for us here starts on page23 , where cosimplicial rings make an appearance.

Morephism appearing in lemma 3.4.2 is the one from the cosimplicial ring of functions on $\mathbf{B}G$ that above I denoted $C^\infty(\mathbf{B}G)$ to the local version, essentially what I denoted $C^\infty_{loc}(\mathbf{B}G)$ (they think algebraically where I had the synthetic smooth version in mind).

Then on the next page, def. 3.4.3 defines the Moore cochain complex associated to a cosimplicial algebra, as well as its normalization. And then prop 3.4.4 is the statement I made above: the normalized Moore cochain complex of the cosimplicial algebra $C^\infty_{loc}(\mathbf{B}G)$ is the Chevalley-Eilenberg algebra of the Lie algebra of $G$ .

All these statements are of course of course “well known”, as they say, in the algebraic case (as David Ben-Zvi has of course amplified here several times in one form or another) and just recalled here.

I am in the process of further collecting this and related material at [[ $\infty$ -quantity]].

Posted by: Urs Schreiber on August 21, 2009 5:12 PM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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Well, what that article does not seem to mention either is the monoidal structure induced on the Chevalley-Eilenberg complex from the cosimplicial ring structure of $C^\infty(\mathbf{B}G)$ …

Posted by: Urs Schreiber on August 21, 2009 5:19 PM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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I started writing a disucssion

Chevalley-Eilenberg algebra in synthetic differential geometry

of how

- starting with the simplicial smooth space $\mathbf{B}G$

- forming in the context of synthetic differential geometry the infinitesimal simplicial space consisting of neighbourhoods of the identities $\mathbf{B} G_e^{(1)}$

- creating from that the cosimplicial algebra of functions $C^\infty(\mathbf{B}G_e^{(1)})$

- then sending this with the monoidal Dold-Kan-correspondence to the differential graded algebra $N^\bullet(C^\infty(\mathbf{B}G_e^{(1)}))$

produces the Chevalley-Eilenberg algebra of $Lie(G)$ :

$N^\bullet(C^\infty(\mathbf{B}G_e^{(1)})) = CE(g) \,.$

This must be well known in some context, but I haven’t seen it discussed in the context of synthetic differential geometry. I’d be grateful for references.

Posted by: Urs Schreiber on August 31, 2009 9:30 PM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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This is not precisely relevant, but you might enjoy the following:

Teleman, Constantin Borel-Weil-Bott theory on the moduli stack of $G$ -bundles over a curve. Invent. Math. 134 (1998), no. 1, 1–57.

in particular the appendices. Other papers of Teleman are also related, in particular you’ll enjoy his preprint with Simpson, De Rham’s theorem for stacks. I learned the simplicial POV on stacks from these papers..and many other things.

Posted by: David Ben-Zvi on August 31, 2009 11:12 PM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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Thanks, David!

It’s late here, so I just looked quickly at the second link that you provided:

you’ll enjoy his preprint with Simpson, De Rham’s theorem for stacks.

Yes, as you can guess, I can relate to that one. Great, thanks. I wasn’t aware of this.

Hm, the proof of prop 3.3, page 7 is not given, is it?

I have my own version of a statement of this sort, but there are some subtle technical differences here which I’d need to think about in order to compare.

I am maybe too tired to seriously read beyond sectin 3. Do the authors produce a full Quillen adjunction inducing the one on the homotopy category?

Posted by: Urs Schreiber on September 1, 2009 12:11 AM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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David Ben-Zvi wrote:

in particular you’ll enjoy his preprint with Simpson, De Rham’s theorem for stacks.

I wrote

I am maybe too tired to seriously read beyond sectin 3.

Now, after a nap or two, I am awake. And I noticed that we talked about this before. So I am posting my reply to this thread here.

Posted by: Urs Schreiber on September 16, 2009 3:04 PM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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One problem induced by the $n$ Lab migration that we are trying to fix is that some long entry names go truncated.

So for the moment the entry announced above is to be reached as

Chevalley-Eilenberg algebra in synthetic differential geomet

no ry

Posted by: Urs Schreiber on September 1, 2009 7:28 PM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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created

$n$ Lab: microlinear space

Posted by: Urs Schreiber on October 6, 2009 9:29 AM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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added

$n$ Lab:tangent bundle - definition in synthetic differential geometry

Posted by: Urs Schreiber on October 6, 2009 10:24 AM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

direct link

(Not that we need it, but isn't it nice that it can be done?)

Posted by: Toby Bartels on October 6, 2009 8:43 PM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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started

$n$ Lab:integration axiom,

but left unfinished, as I am running out of time a bit.

This is supposed to supplement what I am thinking about at

schreiber: $\infty$ -integration axiom.

Posted by: Urs Schreiber on October 6, 2009 11:24 AM | Permalink | Reply to this

Re: Question on Synthetic Differential Forms

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I am thinking about an alternative way of characterizing the infinitesimal simplicial singular complex of a smooth space. I was planning to present this here, but kept being distracted all day by other things.

So all I have now is what I indicate at $n$ Forum: infinitesimal singular simplicial object.

Posted by: Urs Schreiber on October 9, 2009 12:24 AM | Permalink | Reply to this

model structure on dg-algebras

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Does anyone know if there is an electronic copy available or at least some web link for the article

J.F. Jardine, A closed model structure for differential graded algebras , Cyclic Cohomology and Noncommutative Geometry, Fields Institute Communications, Vol. 17, AMS (1997), 55-58.

that is referenced at model structure on dg-algebras?

Posted by: Urs Schreiber on October 27, 2009 12:54 PM | Permalink | Reply to this

Re: model structure on dg-algebras

If you get a hold of a copy, I’d be interested in one as well.

Posted by: Eric Forgy on October 27, 2009 5:28 PM | Permalink | Reply to this

Re: model structure on dg-algebras

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Does anyone know if there is an electronic copy available or at least some web link for the article

J.F. Jardine, A closed model structure for differential graded algebras , Cyclic Cohomology and Noncommutative Geometry, Fields Institute Communications, Vol. 17, AMS (1997), 55-58.

There is now a link here.

Posted by: Urs Schreiber on October 29, 2009 2:10 AM | Permalink | Reply to this

Re: model structure on dg-algebras

Oh my!

Abstract. We derive a closed model structure for the category of noncommutative differential graded algebras over an arbitrary commutative ring with unit.

Those are my favorite kinds :) Thanks for posting the link.

The paper also has all kinds of words familiar from the n-Lab so there is hope I may actually be able to understand some things now :)

Posted by: Eric Forgy on October 29, 2009 8:04 AM | Permalink | Reply to this

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August 3, 2009