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December 25, 2007

This Week’s Finds in Mathematical Physics (Week 260)

Posted by John Baez

In week260 of This Week’s Finds, learn about the Vishniac instability in the Retina Nebula:

Then try my Christmas eve guide to free books on math and physics. There are more and more available! Soon those expensive textbooks will be obsolete. Here’s a nice illustration of Taylor series for the sine function, from Robert Nearing’s online book Mathematical Tools for Physics:

And finally: the ‘exceptional series’ of Lie algebras.

Posted at December 25, 2007 3:15 AM UTC

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Re: This Week’s Finds in Mathematical Physics (Week 260)

Yesterday I bet someone that John would post something here on Christmas day. I won! And a wonderful festive TWF it is.

There are a few more free category theory books in the Reprints series of Theory and Applications of Categories.

Merry Christmas to all!

Posted by: Tom Leinster on December 25, 2007 2:13 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 260)

What did you win? Another present?

I hope you and the Catster are doing well. Haven’t heard much from her lately… I hope that means she’s getting settled down in Sheffield.

Lisa and I are having an extremely peaceful Christmas — just listening to Bach’s Mass in H Minor and eating waffles topped with poppy syrup given to us by that very same Catster. It’s a sunny day here in Riverside, with 50-mile-per-hour gusts knocking down lots of palm fronds into our front yard.

Does anyone here have the energy and smarts to solve this puzzle? I think the answer to question 1 is “2”. The answer to question 2 is “time”. (The remark about Dutch physicists and geese completely gave away the second one.) The answer to question 5 is “1”.

I would keep on slaving away iff 1) I actually liked puzzles, 2) I had the foggiest idea what sort of amusing final answer was supposed to result and 3) I didn’t already have a Perimeter Institute mug.

But, perhaps the nn-Category Café could collectively win this mug! A café should have a bunch of mugs, right?

I hope you’re having fun up there in Glasgow, if that’s where you are now…

Posted by: John Baez on December 25, 2007 5:59 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 260)

Yeah, having fun and eating haggis, exported from Glasgow to Sheffield. Your waffles sound delicious, and I would be jealous… but there is Christmas pudding still to come.

Posted by: Tom Leinster on December 25, 2007 9:05 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 260)

Thank you for the great Christmas present! Merry Christmas!

Posted by: Charlie C on December 25, 2007 4:18 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 260)

You’re welcome! Merry Christmas to you too!

Posted by: John Baez on December 25, 2007 6:00 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 260)

I added a bit of stuff about the “exceptional series” of Lie algebras, since my previous explanation was way too sketchy to be interesting. Now you can see some fun string diagrams!

Posted by: John Baez on December 26, 2007 1:04 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 260)

When you say ‘a Casimir’, do you just mean a central element of the universal enveloping algebra?

Posted by: Greg Muller on December 26, 2007 11:23 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 260)

Yes, ‘a Casimir’ is the same as an element of the center of the universal enveloping algebra of a Lie algebra… though often ‘the Casimir’ means the canonical quadratic element in the center of the universal enveloping algebra of a semisimple Lie algebra.

This is a great example of the mutual unintelligibility of mathematicians and physicists (although you guessed it just fine). Tell mathematicians about a “Casimir” and they’ll usually say “A what?”… and then when you explain it, they’ll say “Why don’t they just call it an element of the center of the universal enveloping algebra?” Tell physicists about an “element of the center of the universal enveloping algebra” and they’ll say “A what?”… and then when you explain it, which can take quite a while if they don’t know about universal enveloping algebras, they’ll say “Why don’t they just call it a Casimir?”

(Physicists can know perfectly well how to work with products of noncommuting Lie algebra elements even without knowing what a ‘universal enveloping algebra’ is. So, if you tell them a ‘Casimir’ is a ‘central element of a universal enveloping algebra’, there’s a good chance they’ll walk away thinking mathematicians get paid by the word.)

Posted by: John Baez on December 27, 2007 3:54 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 260)

Some more links:

This great collection (incl. the famous LNM 900), leading the reader from basic algebra through algebraic number theory, class fields, modular forms, arithmetic groups,… up to etale cohomology, Shimura varieties etc.

Friedhelm Waldhausen’s lectures on algebraic topology and K-theory.

DML: Digital Mathematics Library

G. Harders math-links

MSRI online books

Finally:

“Nearly three and a half centuries of scientific study and achievement is now available online in the Royal Society Journals Digital Archive. This is the longest-running and arguably most influential journal archive in Science, including all the back articles of both Philosophical Transactions and Proceedings.”

Posted by: Thomas Riepe on December 28, 2007 3:38 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 260)

Great stuff, Thomas! If anyone has other favorite free online books, I’d love to see links to them. I think a lot of people keep their books on their personal websites instead of putting them on the arXiv.

Posted by: John Baez on December 30, 2007 1:14 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 260)

I despair of ever keeping up with the energy of the regular members here. However, while organizing some material, I just recalled the discussion on online texts. So I thought I’d post a link to the Southwest Center for Arithmetic Geometry , where I did most of my public service work from 1998 to 2005. If you browse through, you’ll find 10 years worth of surveys, projects, videos, etc. covering virtually all aspects of arithmetic geometry. Consider this part of my feeble attempt to contribute some arithmetic flavor to the diet of the cafe.

MK

Posted by: Minhyong Kim on January 12, 2008 12:20 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 260)

These are not in the mainstream of interests here, but Robert Gray has several free books on his web site. Reflecting his own affiliation, these
books are on topics of interest to mathematically sophisticated electrical engineers, but they’re honest mathematical texts. And in a different direction I’ll second Thomas’s recommendation of Milne’s amazing collection of online course notes.

Posted by: Mark Meckes on January 17, 2008 3:07 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 260)

According to this study of the british library, the neccessary reading skills for making use of such web sources may be rarifying:

“… although young people demonstrate an apparent ease and familiarity with computers, they rely heavily on search engines, view rather than read and do not possess the critical and analytical skills to assess the information… traits that are commonly associated with younger users … are now becoming the norm for all age-groups, from younger pupils and undergraduates through to professors.”

So, like in the middle ages, universities would have to offer courses for learning how to read before trying to use advanced texts in teaching. An other idea were to create software tools for increasing the readability and verificability of mathematical texts, e.g. Dan Grayson’s idea of such a toolbox.

Posted by: Thomas Riepe on January 19, 2008 9:37 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 260)

You mentioned the fact that C(X × Y, Z) ≅ C(X, C(Y, Z)) is not true in general…
I’d like to know more details about this. Is there any book that I can look at?

Posted by: The man in the box on January 7, 2008 10:39 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 260)

Alas, I don’t know counterexamples off the top of my head. You might look at Steenrod’s classic article where he solved this problem by switching to the category of compactly generated Hausdorff spaces. However, even he doesn’t give examples illustrating the problem! He cites an older article by Spanier (ref. 13) which probably does, though.

Someone lurking around here must know the answer to your question. It’s probably in some books, too.

Posted by: John Baez on January 7, 2008 7:27 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 260)

There’s a book called
Counterexamples in topology

part of the issue is what topology to use on C(X,Y)

Posted by: jim stasheff on January 7, 2008 11:22 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 260)

If you restrict to the category of Hausdorff spaces, then I think there’s a theorem that YY is exponentiable iff YY is locally compact. In other words, local compactness is necessary and sufficient for the functor

×Y:HausHaus- \times Y: Haus \to Haus

to possess a right adjoint (which would be this functor ZC(Y,Z)Z \mapsto C(Y, Z)). So in principle, one can always cook up a counterexample by starting with a YY which is not locally compact. (If I remember correctly, the Hausdorffness here can be weakened to sobriety.)

I’d have to dig a little to find a really concrete example. Somehow I think this issue may have arisen before at the Café.

Posted by: Todd Trimble on January 7, 2008 11:52 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 260)

The result about exponentation and local compactness is in Martin Hyland’s Function spaces in the category of locales.

Bas

Posted by: Bas Spitters on January 8, 2008 9:26 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 260)

If anyone is still interested, I’ve managed to find a concrete example of a pair of spaces YY, ZZ for which there is no good topological function space Cont(Y,Z)Cont(Y, Z), i.e., no solution to the universal problem of constructing an isomorphism

hom(X×Y,Z)hom(X,Cont(Y,Z))hom(X \times Y, Z) \cong hom(X, Cont(Y, Z))

natural in spaces XX. (This actually took some doing; it’s based mainly on a close reading of parts of a nice paper by Escardo and Heckmann on exponentiable spaces.)

In our example, YY is the space of rationals and ZZ is the Sierpinski space 2\mathbf{2}, i.e., the set {0,1}\{0, 1\} equipped with the topology consisting of {1}\{1\}, {0,1}\{0, 1\}, and the empty set. In fact any Hausdorff space YY which is not locally compact would do just as well, at the cost of a little extra argumentation.

Suppose (for a contradiction) that there is a topology on the set hom(Y,2)hom(Y, \mathbf{2}) which renders

eval:hom(Y,2)×Y2eval: hom(Y, \mathbf{2}) \times Y \to \mathbf{2}

continuous, and which is universal in the sense that for each continuous map

f:X×Y2f: X \times Y \to \mathbf{2}

there is a unique continuous f^:Xhom(Y,2)\hat{f}: X \to hom(Y, \mathbf{2}) making the obvious triangle commute. Let X=hom(Y,2)X = hom(Y, \mathbf{2}) and consider any topology on XX for which the evaluation map

ev:X×Y2ev: X \times Y \to \mathbf{2}

is continuous; such a topology will be called strong. The corresponding map ev^:Xhom(Y,2)\hat{ev}: X \to hom(Y, \mathbf{2}) is the identity function, and in order for this to be continuous, the putative universal topology must be weaker than the topology on XX. Hence the universal topology, which is itself strong, must be the weakest strong topology (that is, it must equal the intersection of all strong topologies). We will in fact show that this intersection topology cannot possibly be strong. Therefore, no universal strong topology exists.

A continuous map f:Y2f: Y \to \mathbf{2} to Sierpinski space is essentially the same thing as an open set of YY (given by f 1(1)f^{-1}(1)). Therefore hom(Y,2)hom(Y, \mathbf{2}) is essentially the topology Open(Y)Open(Y), and strong topologies amount to topologies on the topology Open(Y)Open(Y), such that the membership relation ε YY×Open(Y)\varepsilon_Y \subseteq Y \times Open(Y) is open.

For example, let COpen(Y)C \subseteq Open(Y) be any collection of open sets. Define a topology T CT_C on Open(Y)Open(Y) to consist of all upward closed sets DOpen(Y)D \subseteq Open(Y) such that CD\union C \in D implies there are finitely many members V 1,,V nV_1, \ldots, V_n of CC such that iV iD\union_i V_i \in D.

Claim: The topology T CT_C on Open(Y)Open(Y) is strong.

Proof: The fact it is a topology is straightforward; we leave this to the reader. To see it is strong, we must show the membership relation ε Y\varepsilon_Y is open in the product space Y×Open(Y)Y \times Open(Y). Take an element (y,V)(y, V) in ε Y\varepsilon_Y (so VV is an open neighborhood of yy); we must find a product neighborhood U×DU \times D of (y,V)(y, V) contained in ε Y\varepsilon_Y. If yy is not contained in C\union C, then put U=VU = V and D={WOpen(Y):VW}D = \{W \in Open(Y): V \subseteq W\}; this DD belongs to T CT_C and xWx \in W whenever xUx \in U and WDW \in D, so U×Dε YU \times D \subseteq \varepsilon_Y. If on the other hand yy belongs to V V^\prime for some V CV^\prime \in C, then put U=VV U = V \cap V^\prime and D={WOpen(Y):VV W}D = \{W \in Open(Y): V \cap V^\prime \subseteq W\}. Again DD belongs to T CT_C and U×Dε YU \times D \subseteq \varepsilon_Y, and we are done.

Since T CT_C is strong, the universal topology is contained in the intersection topology I= COpen(Y)T CI = \bigcap_{C \in Open(Y)} T_C. Therefore (under our supposition), II is strong. This means that for each (y,V)(y, V) such that yVy \in V, there is an open UU and some DID \in I such that yUy \in U, VDV \in D, and U×Dε YU \times D \subseteq \varepsilon_Y. The last condition means UWU \subseteq W for every WDW \in D; in conjunction with the second condition, we see UVU \subseteq V. The second condition also means that whenever COpen(Y)C \subseteq Open(Y) is a covering of VV, some WW in DD is covered by finitely many V 1,,V nV_1, \ldots, V_n in CC (whence the V iV_i form a finite subcover of UWU \subseteq W).

Suppose given such a UU and DD. We are working with the space of rationals YY; choose a rational interval (a,b)(a, b) of yy contained in UU. Choose the cover CC of VV to consist of V(,a)V \cap (-\infty, a), V(b,)V \cap (b, \infty), and the intervals (a+1n,b1n)(a + \frac{1}{n}, b - \frac{1}{n}) [for all nn sufficiently large to make these intervals nonempty]. Since there is no finite subcover of (a,b)(a, b), there can be no finite subcollection of CC which covers UU. This gives the required contradiction.

Posted by: Todd Trimble on January 9, 2008 10:39 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 260)

Todd wrote:

If anyone is still interested…

I am!

Your example implies that the functor ×:TopTop - \times \mathbb{Q}: \mathbf{Top} \to \mathbf{Top} has no right adjoint. A few years ago I tried to find an example of a space YY and a coequalizer not preserved by ×Y- \times Y. I think I’m right in saying that one must exist. The argument was that ×Y- \times Y always preserves sums, so if it preserves coequalizers then it preserves colimits, and therefore (by a result in Borceux’s handbook, in the section on topological function spaces) has a right adjoint. And we know, e.g. by your example, that that needn’t be the case.

This is all getting a bit shaky, but I think it must be the case that there’s a coequalizer not preserved by ×- \times \mathbb{Q}. If it were easy to write one down, that would provide a nice short proof that Top\mathbf{Top} isn’t cartesian closed.

Posted by: Tom Leinster on January 9, 2008 11:14 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 260)

Tom,

While I was digging around, I saw mention several times of this paper by Day and Kelly

  • On topological quotient maps preserved by pullbacks and products, Math. Proc. Cam. Phil. Soc. 67: 553-558, 1970

which apparently is well-known to experts in this area and which I think is relevant to your idea (although I haven’t seen that paper). Escardo and Heckmann mention it in the paper I linked to above.

It would be great if you cooked up something simpler than what I wrote (which took me hours to hack out!), based on your idea.

Posted by: Todd Trimble on January 9, 2008 11:38 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 260)

Thanks, Todd. When I was thinking about this before, I got some very helpful mails from Martín Escardó, who also mentioned the Day–Kelly paper. Apparently they show in it that the following conditions on a topological space YY are equivalent:

  • YY is exponentiable (i.e. ×Y- \times Y has a right adjoint)
  • ×Y- \times Y preserves quotient maps (i.e. if q:ABq: A \to B is a quotient map then so is q×1 Y:A×YB×Yq \times 1_Y: A \times Y \to B \times Y)
  • the ordered set of open subsets of YY is a continuous lattice (in the sense of Scott).

Probably guessing that I didn’t know what a continuous lattice was, Martín also pointed out that for Hausdorff spaces, the last condition is equivalent to local compactness. So, any Hausdorff non-locally-compact space YY, such as \mathbb{Q}, has the property that ×Y- \times Y doesn’t preserve quotient maps. And quotient maps are coequalizers, so this confirms my suspicions.

Of course, this still doesn’t give us a particular coequalizer/quotient that fails to be preserved by ×- \times \mathbb{Q}. Maybe a close reading of Day and Kelly would yield one… but I think I’m done on this one, for now at least.

Posted by: Tom Leinster on January 10, 2008 4:25 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 260)

Hi Tom –

I ran across this query of yours from two and a half years ago (!), and decided to give it another crack. You wrote

[Any] Hausdorff non-locally-compact space YY, such as \mathbb{Q}, has the property that ×Y− \times Y doesn’t preserve quotient maps. And quotient maps are coequalizers, so this confirms my suspicions. Of course, this still doesn’t give us a particular coequalizer/quotient that fails to be preserved by − \times \mathbb{Q}. Maybe a close reading of Day and Kelly would yield one…

I never did get hold of Day & Kelly, but I think what I wrote earlier can be repackaged to yield a specific example. Not an easy example unfortunately – I wish I knew one.

Let YY be Hausdorff but not locally compact. To give a coequalizer not preserved by the functor Y×Y \times -, it suffices to give an example of a colimit not preserved by this functor. (For we can present the colimit as a coequalizer of maps between coproducts, and since Y×Y \times - preserves coproducts, it must not preserve this coequalizer.)

Let 2\mathbf{2} be Sierpinski space: the set 2={0,1}2 = \{0, 1\} where {1}\{1\} is open but {0}\{0\} is not. Let 𝒪(Y)\mathcal{O}(Y) denote the topology of YY, and let

char:Y×𝒪(Y)2char: Y \times \mathcal{O}(Y) \to \mathbf{2}

be the characteristic function, taking a pair (y,V)(y, V) to 11 precisely when yVy \in V. Now consider the diagram in TopTop whose nodes are all topological space structures on 𝒪(Y)\mathcal{O}(Y) which render charchar continuous (call such topologies 𝒯\mathcal{T} on 𝒪(Y)\mathcal{O}(Y) strong), and whose arrows are continuous identity functions

id:𝒪(Y) 𝒯𝒪(Y) 𝒯,id: \mathcal{O}(Y)_{\mathcal{T}} \to \mathcal{O}(Y)_{\mathcal{T}'},

meaning there is an inclusion 𝒯𝒯\mathcal{T}' \hookrightarrow \mathcal{T} of strong topologies. I claim Y×Y \times - cannot preserve the colimit of this (directed) diagram.

The colimit in TopTop of this diagram is 𝒪(Y)\mathcal{O}(Y) topologized by the intersection of all strong topologies. If Y×Y \times - preserved this colimit, then we would have an induced continuous map

char:Y×colim 𝒯𝒪(Y) 𝒯2char: Y \times colim_{\mathcal{T}} \mathcal{O}(Y)_{\mathcal{T}} \to \mathbf{2}

so that the intersection of all strong topologies would itself be strong. I will show this is impossible, by showing that for every strong topology 𝒯\mathcal{T}, there is a strictly smaller strong topology 𝒯𝒯\mathcal{T}' \hookrightarrow \mathcal{T}. This will rely on the non-local-compactness of YY. Specifically, we use the fact that if YY is Hausdorff but not locally compact, then we can choose an open VV and a point yVy \in V such that for all open UU with yUVy \in U \subseteq V, there exists a non-compact closed CC with yCUy \in C \subseteq U.

A topology 𝒯\mathcal{T} is strong iff charchar is continuous iff the elementhood relation Y=char 1(1)\in_Y = char^{-1}(1) is open in Y×𝒪(Y) 𝒯Y \times \mathcal{O}(Y)_{\mathcal{T}}. In that case, the chosen point (y,V)(y, V) belongs to Y\in_Y (i.e., yVy \in V), so there exist open sets UYU \subseteq Y, D𝒯D \in \mathcal{T}, such that

  1. yUy \in U
  2. VDV \in D
  3. U×D YU \times D \subseteq \in_Y

Lemma: The third condition is equivalent to the condition

UD={W𝒪(Y):WD}.U \subseteq \bigcap D = \bigcap \{W \in \mathcal{O}(Y): W \in D\}.

Together with conditions 1 and 2, this implies yUVy \in U \subseteq V.

We can find a non-compact closed set CUC \subseteq U containing yy. Let U aU_a be an open cover of CC with no finite subcover. There is no harm in assuming that the difference VCV - C is one of the U aU_a.

Now comes a trick^\dagger (which I won’t try to motivate). Define a new topology 𝒰\mathcal{U} to consist of those D𝒪(Y)D' \subseteq \mathcal{O}(Y) such that

  • DD' is upward-closed: if VDV \in D' and VVV \subseteq V' is an inclusion of open sets of YY, then VDV' \in D'.
  • If the union of all the U aU_a belongs to DD', then some finite union U a 1U a nU_{a_1} \cup \ldots \cup U_{a_n} belongs to DD'.

It is not hard to check that 𝒰\mathcal{U} is indeed a topology. To check that it is strong, we need to check that the membership relation Y\in_Y is open in Y×𝒪(Y) 𝒰Y \times \mathcal{O}(Y)_{\mathcal{U}}. Suppose (x,W) Y(x, W) \in \in_Y, i.e., suppose xWx \in W. If xx does not belong to the union of all the U aU_a, then

W×{W:WW}W \times \{W': W \subseteq W'\}

is a 𝒰\mathcal{U}-open neighborhood of (x,W)(x, W) included in Y\in_Y (by the lemma above). If on the other hand xx belongs to some U aU_a, then

(WU a)×{W:WU aW}(W \cap U_a) \times \{W': W \cap U_a \subseteq W'\}

is a 𝒰\mathcal{U}-open neighborhood of (x,W)(x, W), which (again by the lemma) is included in Y\in_Y.

However, the D𝒯D \in \mathcal{T} with which we started the argument does not belong to 𝒰\mathcal{U}. Suppose otherwise. We know that the union of the U aU_a contains VV and VDV \in D, so aU a\bigcup_a U_a belongs to DD by upward closure. Thus some finite union U a 1U a nU_{a_1} \cup \ldots \cup U_{a_n} would belong to DD. But any member of DD contains UU, hence we have

CUU a 1U a nC \subseteq U \subseteq U_{a_1} \cup \ldots \cup U_{a_n}

contradicting the fact that there is no finite subcover of CC.

Therefore the topology 𝒯=𝒯𝒰\mathcal{T}' = \mathcal{T} \cap \mathcal{U} is strictly smaller than 𝒯\mathcal{T}. The intersection 𝒯\mathcal{T}' of two strong topologies is also strong, so we are done.

^\dagger The unmotivated trick we used above is closely related to the so-called Scott topology on a poset.

Posted by: Todd Trimble on June 20, 2010 3:23 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 260)

There is another neat example (due to Kathleen Lewis) in 1.7.1 of May-Sigurdsson: ×\mathbb{Q}\times - does not preserve the quotient ×\mathbb{Q}\times\mathbb{N} \to \mathbb{Q} \wedge \mathbb{N}, and therefore the smash product of ordinary topological spaces is not associative: ()()\mathbb{Q} \wedge (\mathbb{Q}\wedge \mathbb{N}) \ncong (\mathbb{Q}\wedge \mathbb{Q}) \wedge \mathbb{N}. I haven’t read through the details of the argument, though.

Posted by: Mike Shulman on June 20, 2010 5:08 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 260)

That sounds promising, thanks! But for some reason I wasn’t able to download the .pdf when I click on the link (nor was I able to when I went directly to May’s website).

Posted by: Todd Trimble on June 20, 2010 6:07 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 260)

The link is working fine for me now.

Posted by: John Baez on June 21, 2010 7:40 PM | Permalink | Reply to this

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