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June 6, 2007

Cohomology and Computation (Week 25)

Posted by John Baez

In this week’s seminar on Cohomology and Computation, we put the pieces together and saw how to build simplicial objects from adjoint functors. Next time we’ll see what they’re actually like, in an example:

  • Week 25 (May 24) - The bar construction, continued. The zig-zag identities for the unit and counit of an adjunction. Monads and comonads from adjunctions. Simplicial objects from adjunctions.

Last week’s notes are here; next week’s notes are here.

Posted at June 6, 2007 5:09 PM UTC

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12 Comments & 3 Trackbacks

Read the post Cohomology and Computation (Week 24)
Weblog: The n-Category Café
Excerpt: What makes the bar construction tick?
Tracked: June 6, 2007 5:21 PM

Re: Cohomology and Computation (Week 25)

Might Δ\Delta have a Leinster-Euler cardinality, using trickery if necessary? How many order preserving maps from [mm] to [nn] are there?

Posted by: David Corfield on June 7, 2007 4:07 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 25)

How many order preserving maps from [m][m] to [n][n] are there?

This many (A071919). I like the way it unfolds just like Pascal’s triangle:

Let’s write (m,n)(m,n) for the number of order-preserving maps [m][n][m]\to[n]. Now, a map [m+1][n+1][m+1]\to[n+1] either maps the top element of [m+1][m+1] to the top element of [n+1][n+1] – there are (m,n+1)(m,n+1) of these – or it doesn’t, in which case nothing can be mapped to the top element, and there are (m+1,n)(m+1, n) of those. So

(1)(m+1,n+1)=(m,n+1)+(m+1,n).(m+1, n+1) = (m,n+1) + (m+1, n).
Posted by: Robin on June 7, 2007 6:26 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 25)

Shouldn’t that be (m+1,n+1)=(m,n)+(m+1,n)?

Either nothing goes to the top element [n+1] so all m+1 have to go into [n], or the top element goes to the top element so the remaining m elements have to go into [n]. Or have I missed something?

Posted by: John Armstrong on June 7, 2007 6:45 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 25)

You may have been thinking of maps which preserve the relation “less than” instead of “less than or equal to”. In other words, you are counting the injective order-preserving maps from [m] to [n].

Posted by: Todd Trimble on June 7, 2007 6:56 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 25)

Ah, that does it.

Posted by: John Armstrong on June 7, 2007 7:13 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 25)

Thanks. There’s an initial object so my question was rather unnecessary, and the Leinster-Euler cardinality is 1, fitting well with the sense that Δ\Delta captures the interval.

Actually what I was really interested in was the cyclic category, often denoted Λ\Lambda, introduced by Connes to capture the action of a circle combinatorially. Λ\Lambda is the category whose objects [nn] are circles marked by the sets of n thn^{th} roots of unity. Maps are degree 1 mappings between circles which send marked points to marked points, and are increasing. This is quite like Δ\Delta but where you are allowed to cycle round the domain before you choose an order-preserving map.

Λ\Lambda has the pleasant feature of being isomorphic to its opposite. You can count its number of morphisms by multiplying each of the rows of Pascal’s triangle by the row number and twisting it anti/counter-clockwise into a square matrix.

(1)(1 2 3 4 ... 2 6 12 20 ... 3 12 30 60 ... 4 20 60 140 ... ... ... ... ... ...) \left( \array{1 & 2 & 3 & 4 & ... \\ 2 & 6 & 12 & 20 & ... \\ 3 & 12 & 30 & 60 & ...\\ 4 & 20 & 60 & 140 & ...\\ ... & ... & ... & ... & ...} \right)

From what I can see, if you try to calculate the Leinster-Euler characteristic for this category restricted to the first nn objects, you get the harmonic series up to 1/nn.

This would suggest that the L-E characteristic is infinite. A good thing because BΛ=BS 1=P ()B\Lambda = B S^1 = P_\infty(\mathbb{C}), and if the cardinality of the circle is 0, we’d want the cardinality of Λ\Lambda to be 10\frac{1}{0}.

Posted by: David Corfield on June 8, 2007 7:45 AM | Permalink | Reply to this

Re: Cohomology and Computation (Week 25)

There they are – denominators in Leibniz’s Harmonic Triangle.

Posted by: David Corfield on June 8, 2007 12:45 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 25)

Cool! I’d never heard of Leibniz’s harmonic triangle before. In an amusing parody of Pascal’s triangle, each entry is the sum of the two below it!

One thing that instantly struck me here is the number 42. This is the number of vertices of the 4-dimensional associahedron. In other words, it’s the number of ways to parenthesize a product of 6 things. That’s because 42 is the 5th Catalan number:

1,2,5,14,42,132,429,...1, 2, 5, 14, 42, 132, 429, ...

(It should be the 6th, but someone started counting them wrong, and it may be too late to change the convention here.)

Indeed, I’ve never seen the number 42 before in any interesting role that didn’t arise from the Catalan numbers!

However, the Catalan connection seems to be a coincidence this time. The Catalan numbers are

1n+1((2nn)), \frac{1}{n+1} \left(\binom{2n}{n}\right) ,

while the Leibniz triangle, as you mention, consists of numbers

1k((nk)). \frac{1}{k \left(\binom{n}{k}\right)}.

I was also struck by the number 105, which I’d only seen before in certain Feynman diagram calculations, basically because

x 8e x 2/2dx=1052π \int_{-\infty}^\infty x^8 e^{-x^2/2} d x = 105 \sqrt{2 \pi }

which ultimately comes from

7!!=1357=105.7!! = 1 \cdot 3 \cdot 5 \cdot 7 = 105.

In other words, there are 105 ways to pair up 8 people. The integral above can be done using this fact and some stuff about annihilation and creation operators.

But again, this seems to be a coincidence!

So, my usual pattern-recognition algorithms are just turning up noise. We need to ponder the combinatorics lurking behind the cyclic category using some other ideas…

For starters, why did Leibniz invent this “harmonic triangle”? What’s it good for?

Posted by: John Baez on June 9, 2007 6:54 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 25)

I was also struck by the number 105, which I’d only seen before in certain Feynman diagram calculations

This is probably just silly, but the first thing 105 trips in my memory is that it’s the order of the operation UR on Rubik’s supercube. Mark the orientation of each face cubie and then start twisting the top clockwise by a quarter turn, then the right clockwise by a quarter turn, and so on. After 105 pairs of twists you’re back where you started.

Posted by: John Armstrong on June 9, 2007 7:33 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 25)

Bah, I got this wrong.

105 is the order of that operation on the Magic Cube. On the Supercube (UR)^{105} twists the upper and right face cubies by a quarter-turn each. Ern? is surely disappointed in me :(

Posted by: John Armstrong on June 9, 2007 9:32 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 25)

I pointed out that 105 is the double factorial 7!!=13577!! = 1 \cdot 3 \cdot 5 \cdot 7, while 42 is the 5th Catalan number.

I just noticed an amusing relation between double factorials and Catalan numbers, quite irrelevant to this thread.

The double factorial (2n1)!!(2n-1)!! counts how many ways 2n2n people can pair off into couples. The Catalan number C nC_n counts how many ways these people sitting around a circular table can reach out with their right hand and touch a partner’s right hand without any arms crossing.

For example, when n=2n = 2, we have 3 pairings of 4 people, but only 2 noncrossing pairings. For n=3n = 3 we have 15 pairings of 6 people, but only 5 noncrosssing pairings. For n=4n = 4 we have 105 pairings of 8 people, but only 14 noncrossing pairings. And so on.

Numerology is a dangerous activity which should be left to trained mathematicians. I took a chance and looked at the Wikipedia article on the number 42. I learned that 10 factorial seconds is exactly

days.

Another useless fact to clog my brain.

Posted by: John Baez on June 9, 2007 9:15 PM | Permalink | Reply to this

Re: Cohomology and Computation (Week 25)

Apparently, Leibniz discussed the harmonic triangle in “Historia and Origo Calculi Differentialis” (1714), published in 1849.

Also from that mailing list,

As far as I’ve read, Niman suggests convincingly that Leibniz devised his triangle in attempting to find the sum of the infinite series whose terms are the reciprocal triangular numbers, a problem posed to Leibniz by Huygens in 1672.

Here (p. 3) it is suggested that it played an important part in Leibniz’s creation of the calculus.

Elsewhere, we find that entries correspond to the values of the beta function.

Posted by: David Corfield on June 10, 2007 11:42 AM | Permalink | Reply to this
Read the post The Curious Incident of the Dog in the Night-time
Weblog: The n-Category Café
Excerpt: The cyclic category
Tracked: June 8, 2007 8:18 AM
Read the post Cohomology and Computation (Week 26)
Weblog: The n-Category Café
Excerpt: An example of the bar construction: puffing up a point to the free contractible G-space EG, important in group cohomology.
Tracked: June 18, 2007 7:18 AM

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