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am giving this its own page, for ease of hyperlinking examples relevant at all of Cayley graph, Cayley distance, Cayley distance kernel, kernel method and maybe elsewhere
for completeness, to go alongside Mallows kernel
for completeness, to go alongside Kendall distance
Created with the following content:
A module over a commutative ring is dualizable if it is a dualizable object in the symmetric monoidal category of -modules equipped with the tensor product over .
Since this [symmetric monoidal category is a closed monoidal category, the dual object to is necessarily .
Furthermore, the abstract evaluation map
must coincide with the map induced by the bilinear map
that sends to .
\begin{theorem} An -module is dualizable if and only if it is a finitely generated projective module. \end{theorem}
\begin{proof} First, dualizable objects are closed under retracts and finite direct sums. Any finitely generated projective module is a retract of for some , so to show that finitely generated projective modules are dualizable, it suffices to observe that is dualizable as an -module.
Conversely, we show that dualizable objects are finitely generated projective modules. Unfolding the definition of a dualizable object, an -module is dualizable if the coevaluation map
and the evaluation map
satisfy the triangle identities:
The coevaluation map sends to a finite sum
The triangle identities now read
The first identity implies that () generate as an -module, i.e., is finitely generated.
Consider the map that sends to . Consider also the map that sends to . The first triangle identity now reads . Thus, is a retract of , i.e., is a projective module. \end{proof}
a stub (though I did try my hand on a brief idea-section), for the moment mostly to provide a home for
I have added to M5-brane a fairly detailed discussion of the issue with the fractional quadratic form on differential cohomology for the dual 7d-Chern-Simons theory action (from Witten (1996) with help of Hopkins-Singer (2005)).
In the new section Conformal blocks and 7d Chern-Simons dual.
added a bunch of references to M2-brane
brief category:people-entry for hyperlinking references at kernel methods and at machine learning
I have added DOI and author link to:
I was going to ask if this is the same “Phil Hanlon” as in
but I suppose now it must be so
added a bit more text to the Idea-section at Wick rotation and in particular added cross-links with Osterwalder-Schrader theorem.
brief category:people-entry for hyperlinking references at Cayley graph
brief category:people-entry for hyperlinking references at n-Lie algebra
brief category:people-entry for hyperlinking references at n-Lie algebra and at geometric group theory, specifically Cayley distance
(I hope I am identifying this author correctly, matching the Wikipedia entry to the cited references. If anyone knows more, please double-check.)
brief category:people-entry for hyperlinking references at confinement
stub for confinement, but nothing much there yet. Just wanted to record the last references there somewhere.
created worldline formalism to go with this Physics.SE answer
added some formatting and some cross-links (nilpotent groups!) and added pointer to:
brief category:people-entry for hyperlinking references at Cayley distance, Mallows kernel etc.
brief category:people-entry for hyperlinking references at Cayley distance and Mallows kernel etc.
giving cycle of a permutation its own hyperlink
cross-linked with Cayley graph and added this pointer:
brief category:people-entry for hyperlinking references at Mallows kernel
starting something, mainly to have a place to record
brief category:people-entry for hyperlinking references at Kendall tau distance
I’ve expanded the section on morphisms in Banach space, because the new page on isomorphism classes of Banach spaces refers to a different notion of isomorphism than what the Banach space page previously called the “usual” notion of isomorphism. (The issue is that what’s usual seems to be different for analysts and category theorists.)
brief category:people-entry for hyperlinking references at Mallows kernel
brief category:people-entry for hyperlinking references at Mallows kernel
a stub, for the moment just so as to record pointer to Simpson 12 where “resolution of the paradox” is claimed to be achieved simply by passing from topological spaces to locales
some minimum, just for completeness for links at Cayley graph, finitely generated group and geometric group theory
some minimum, to connect to Cayley graph and geometric group theory
brief category:people-entry for hyperlinking references at geometric group theory
brief category:people-entry for hyperlinking references at geometric group theory
reformatted the entry group a little, expanded the Examples-section a little and then pasted in the group-related “counterexamples” from counterexamples in algebra. Mainly to indicate how I think this latter entry should eventually be used to improve the entries that it refers to.
while bringing some more structure into the section-outline at comma category I noticed the following old discussion there, which hereby I am moving from there to here:
[begin forwarded discussion]
+–{.query} It's a very natural notation, as it generalises the notation (or as is now more common) for a hom-set. But personally, I like (or if you want to differentiate from a cocomma category, but that seems an unlikely confusion), as it is a category of arrows from to . —Toby Bartels
Mike: Perhaps. I never write for a hom-set, only or where is the category involved, and this is also the common practice in nearly all mathematics I have read. I have seen for an internal-hom object in a closed monoidal category, and for a hom-set in a homotopy category, but not for a hom-set in an arbitrary category.
I would be okay with calling the comma category (or more generally the comma object) or if you are considering it as a discrete fibration from to . But if you are considering it as a category in its own right, I think that such notation is confusing. I don’t mind the arrow notations, but I prefer as less visually distracting, and evidently a generalization of the common notation for a slice category.
Toby: Well, I never stick ‘’ in there unless necessary to avoid ambiguity. I agree that the slice-generalising notation is also good. I'll use it too, but I edited the text to not denigrate the hom-set generalising notation so much.
Mike: The main reason I don’t like unadorned for either comma objects or hom-sets is that it’s already such an overloaded notation. My first thought when I see in a category is that we have and and we’re talking about the pair — surely also a natural generalization of the very well-established notation for ordered pairs.
Toby: The notation for a double comma object makes me like even more!
Mike: I’d rather avoid using in the name of an object; talking about projections looks a good deal more confusing to me than .
Toby: I can handle that, but after thinking about it more, I've realised that the arrow doesn't really work. If , then ought to be the set of transformations between them. (Or , but you can't keep that decoration up.)
Mike: Let me summarize this discussion so far, and try to get some other people into it. So far the only argument I have heard in favor of the notation is that it generalizes a notation for hom-sets. In my experience that notation for hom-sets is rare-to-nonexistent, nor do I like it as a notation for hom-sets: for one thing it doesn’t indicate the category in question, and for another it looks like an ordered pair. The notation for a comma category also looks like an ordered pair, which it isn’t. I also don’t think that a comma category is very much like a hom-set; it happens to be a hom-set when the domains of and are the point, but in general it seems to me that a more natural notion of hom-set between functors is a set of natural transformations. It’s really the fibers of the comma category, considered as a fibration from to , that are hom-sets. Finally, I don’t think the notation scales well to double comma objects; we could write but it is now even less like a hom-set.
Urs: to be frank, I used it without thinking much about it. Which of the other two is your favorite? By the way, Kashiwara-Schapira use . Maybe ? Lengthy, but at least unambiguous. Or maybe ?
Zoran Skoda: or are the only two standard notations nowdays, I think the original which was done for typographical reasons in archaic period is abandonded by the LaTeX era. is more popular among practical mathematicians, and special cases, like when ) and among category experts…other possibilities for notation should be avoided I think.
Urs: sounds good. I’ll try to stick to then.
Mike: There are many category theorists who write , including (in my experience) most Australians. I prefer myself, although I occasionally write if I’m talking to someone who I worry might be confused by .
Urs: recently in a talk when an over-category appeared as somebody in the audience asked: “What’s that quotient?”. But already looks different. And of course the proper even more so.
Anyway, that just to say: i like , find it less cumbersome than and apologize for having written so often.
Toby: I find more self explanatory, but is cool. was reasonable, but we now have better options.
=–
Created:
Anton Suschkewitsch (Антон Казимирович Сушкевич) was a Russian mathematician working on semigroup theory.
I have begun cleaning up the entry cycle category, tightening up definitions and proofs. This should render some of the past discussion obsolete, by re-expressing the intended homotopical intuitions (in terms of degree one maps on the circle) more precisely, in terms of “spiraling” adjoints on the poset .
Here is some of the past discussion I’m now exporting to the nForum:
The cycle category may be defined as the subcategory of Cat whose objects are the categories which are freely generated by the graph , and whose morphisms are precisely the functors of degree (seen either at the level of nerves or via the embedding given by on the level of objects, the rest being obvious).
The simplex category can be identified with a subcategory of , having the same objects but with fewer morphisms. This identification does not respect the inclusions into , however, since and are different categories.
I left a counter-query underneath Zoran’s query at compactly generated space. It may be time for a clean-up of this article; the query boxes have been left dangling and unanswered for quite some time. Either proofs or references to detailed proofs would be welcome.
for completeness, with pointer to