Comments for blog :: Brent -> [String]
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Comment on Monoidal sparks by L Spice
https://byorgey.wordpress.com/2018/10/01/monoidal-sparks/#comment-19630
Tue, 06 Nov 2018 00:03:09 +0000http://byorgey.wordpress.com/?p=2159#comment-19630As I mentioned https://byorgey.wordpress.com/2018/10/01/monoidal-sparks/#comment-19423 , you get the 3D Heisenberg group over ℝ.
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Comment on Catsters guide by Brent
https://byorgey.wordpress.com/catsters-guide-2/#comment-19474
Tue, 16 Oct 2018 02:40:28 +0000http://byorgey.wordpress.com/?page_id=1116#comment-19474Ah, thanks! But hmm, Eckmann-Hilton seems to mention monoid objects a bunch. Have we discovered a cycle?
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Comment on Monoidal sparks by Brent
https://byorgey.wordpress.com/2018/10/01/monoidal-sparks/#comment-19460
Mon, 15 Oct 2018 11:12:27 +0000http://byorgey.wordpress.com/?p=2159#comment-19460Oh, interesting example! Off the top of my head I can’t seem to make any sense of it geometrically though.
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Comment on Monoidal sparks by Philipp Schuster
https://byorgey.wordpress.com/2018/10/01/monoidal-sparks/#comment-19458
Mon, 15 Oct 2018 10:21:53 +0000http://byorgey.wordpress.com/?p=2159#comment-19458Perhaps another example: Let A be the two-dimensional vector space over the reals with vector addition and let B be the real numbers with addition. Let “spark” be the dot product. What does your proposed combining operation mean geometrically?
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Comment on Catsters guide by Bartosz Milewski
https://byorgey.wordpress.com/catsters-guide-2/#comment-19440
Thu, 11 Oct 2018 13:42:29 +0000http://byorgey.wordpress.com/?page_id=1116#comment-19440I would suggest moving Monoid objects down in the topological sort, because Monoid object 2 depends on Eckmann-Hilton.
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Comment on Monoidal sparks by Brent
https://byorgey.wordpress.com/2018/10/01/monoidal-sparks/#comment-19435
Thu, 11 Oct 2018 02:12:02 +0000http://byorgey.wordpress.com/?p=2159#comment-19435What a fascinating idea. ;)
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Comment on Monoidal sparks by Edward Kmett
https://byorgey.wordpress.com/2018/10/01/monoidal-sparks/#comment-19429
Wed, 10 Oct 2018 04:09:59 +0000http://byorgey.wordpress.com/?p=2159#comment-19429It is almost like this might be a useful building block for counting the number of inversions in an array or something. ;)
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Comment on Monoidal sparks by L Spice
https://byorgey.wordpress.com/2018/10/01/monoidal-sparks/#comment-19423
Tue, 09 Oct 2018 14:43:39 +0000http://byorgey.wordpress.com/?p=2159#comment-19423This is very close to the construction of the Heisenberg group, for which $A$ is a symplectic space over an odd-characteristic field $B$, and the ‘spark’ is the symplectic pairing. In the case where $A \cong V \oplus V^*$, where $V$ is a vector space over $B$, with its natural symplectic pairing, one can realise the Heisenberg group as a group of upper unitriangular $(2n + 1)$-square matrices (where $n$ is the dimension of $V$). One can imagine a generalisation of this construction involving a matrix of (commutative, to be safe) monoids $M_{i j}$ and, for each tuple $(i, j, k)$, a ‘spark’ $M_{i j} \otimes M_{j k} \to M_{i k}$.
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Comment on Monoidal sparks by Brent
https://byorgey.wordpress.com/2018/10/01/monoidal-sparks/#comment-19415
Mon, 08 Oct 2018 22:30:31 +0000http://byorgey.wordpress.com/?p=2159#comment-19415No worries! =)
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Comment on Monoidal sparks by Yitz
https://byorgey.wordpress.com/2018/10/01/monoidal-sparks/#comment-19410
Mon, 08 Oct 2018 14:28:28 +0000http://byorgey.wordpress.com/?p=2159#comment-19410Ah I see, you also wrote

> Similar laws should hold if we fix the second argument of – \cdot – instead of the first.