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- Discussion Type
- discussion topicCayley graph of Sym(3)
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active 4 hours ago

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

- Discussion Type
- discussion topichorizontal chord diagram
- Category Latest Changes
- Started by Urs
- Comments 63
- Last comment by Urs
- Last Active 4 hours ago

- Discussion Type
- discussion topicsymmetric group
- Category Latest Changes
- Started by Tim_Porter
- Comments 8
- Last comment by Urs
- Last Active 4 hours ago

- Discussion Type
- discussion topicCayley distance kernel
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 4 hours ago

for completeness, to go alongside

*Mallows kernel*

- Discussion Type
- discussion topicCayley distance
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active 5 hours ago

for completeness, to go alongside

*Kendall distance*

- Discussion Type
- discussion topicCayley graph
- Category Latest Changes
- Started by Tim_Porter
- Comments 7
- Last comment by Urs
- Last Active 5 hours ago

- Discussion Type
- discussion topicdualizable module
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 4
- Last comment by Dmitri Pavlov
- Last Active 5 hours ago

Created with the following content:

## Idea

## Definition

A module $M$ over a commutative ring $R$ is

**dualizable**if it is a dualizable object in the symmetric monoidal category of $R$-modules equipped with the tensor product over $R$.Since this [symmetric monoidal category is a closed monoidal category, the dual object to $M$ is necessarily $Hom_R(M,R)$.

Furthermore, the abstract evaluation map

$Hom_R(M,R)\otimes_R M\to R$must coincide with the map induced by the bilinear map

$Hom_R(M,R)\times_R M\to R$that sends $(f,m)$ to $f(m)$.

## Characterization

\begin{theorem} An $R$-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 $R^n$ for some $n\ge0$, so to show that finitely generated projective modules are dualizable, it suffices to observe that $R$ is dualizable as an $R$-module.

Conversely, we show that dualizable objects are finitely generated projective modules. Unfolding the definition of a dualizable object, an $R$-module $M$ is dualizable if the coevaluation map

$coev: R \to M\otimes Hom_R(M,R)$and the evaluation map

$ev: Hom_R(M,R)\otimes M\to R$satisfy the triangle identities:

$(id_M \otimes ev)\circ (coev\otimes id_M) = id_M,$ $(ev \otimes id_{Hom(M,R)})\circ (id_{Hom(M,R)}\otimes coev) = id_{Hom(M,R)}.$The coevaluation map sends $1\in R$ to a

$\sum_{i\in I} m_i\otimes f_i.$*finite*sumThe triangle identities now read

$\sum_{i\in I} m_i f_i(p) = p,\qquad p\in M$ $\sum_{i\in I} r(m_i) f_i = r, \qquad r\in Hom_R(M,R).$The first identity implies that $m_i$ ($i\in I$) generate $M$ as an $R$-module, i.e., $M$ is finitely generated.

Consider the map $a: R^I\to M$ that sends $(r_i)_{i\in I}$ to $\sum_{i\in I} m_i r_i$. Consider also the map $b: M\to R^I$ that sends $p\in M$ to $(f_i(p))_{i\in I}\in R^I$. The first triangle identity now reads $b a = id_M$. Thus, $M$ is a retract of $R^I$, i.e., $M$ is a projective module. \end{proof}

## Related concepts

- Discussion Type
- discussion topicgauge theory
- Category Latest Changes
- Started by nLab edit announcer
- Comments 4
- Last comment by nLab edit announcer
- Last Active 6 hours ago

- Discussion Type
- discussion topickernel method
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by David_Corfield
- Last Active 11 hours ago

a stub (though I did try my hand on a brief idea-section), for the moment mostly to provide a home for

- Julien Mairal, Jean-Philippe Vert,
*Machine Learning with Kernel Methods*, 2017 (pdf)

- Julien Mairal, Jean-Philippe Vert,

- Discussion Type
- discussion topicPichet Vanichchapongjaroen
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 13 hours ago

- Discussion Type
- discussion topicM5-brane
- Category Latest Changes
- Started by Urs
- Comments 22
- Last comment by Urs
- Last Active 13 hours ago

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*.

- Discussion Type
- discussion topicM2-brane
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active 13 hours ago

added a bunch of references to

*M2-brane*

- Discussion Type
- discussion topicJulien Mairal
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 13 hours ago

brief

`category:people`

-entry for hyperlinking references at*kernel methods*and at*machine learning*

- Discussion Type
- discussion topicmachine learning
- Category Latest Changes
- Started by David_Corfield
- Comments 2
- Last comment by Urs
- Last Active 14 hours ago

- Discussion Type
- discussion topicbidirectional typechecking
- Category Latest Changes
- Started by Mike Shulman
- Comments 46
- Last comment by nLab edit announcer
- Last Active 18 hours ago

- Discussion Type
- discussion topicn-Lie algebra
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Dmitri Pavlov
- Last Active 23 hours ago

I have added DOI and author link to:

- Phil Hanlon, Michelle Wachs,
*On Lie $k$-Algebras*, Advances in Mathematics Volume 113, Issue 2, July 1995, Pages 206–236 (doi:10.1006/aima.1995.1038)

I was going to ask if this is the same “Phil Hanlon” as in

- Persi Diaconis, Phil Hanlon,
*Eigen Analysis for Some Examples of the Metropolis Algorithm*, in Donald Richards (ed.)*Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications*, Contemporary Mathematics Vol. 138, AMS 1992 (doi:10.1090/conm/138, EFS NSF 392, pdf)

but I suppose now it must be so

- Phil Hanlon, Michelle Wachs,

- Discussion Type
- discussion topicmodel structure on an over category
- Category Latest Changes
- Started by Urs
- Comments 23
- Last comment by Mike Shulman
- Last Active 1 day ago

- Discussion Type
- discussion topicWick rotation
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active 1 day ago

added a bit more text to the Idea-section at

*Wick rotation*and in particular added cross-links with*Osterwalder-Schrader theorem*.

- Discussion Type
- discussion topicElena Konstantinova
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

brief

`category:people`

-entry for hyperlinking references at*Cayley graph*

- Discussion Type
- discussion topicMichelle Wachs
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

brief

`category:people`

-entry for hyperlinking references at*n-Lie algebra*

- Discussion Type
- discussion topicPhil Hanlon
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

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.)

- Discussion Type
- discussion topicYurii Antonovich Simonov
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 1 day ago

brief

`category:people`

-entry for hyperlinking references at*confinement*

- Discussion Type
- discussion topicconfinement
- Category Latest Changes
- Started by Urs
- Comments 31
- Last comment by Urs
- Last Active 1 day ago

stub for

*confinement*, but nothing much there yet. Just wanted to record the last references there somewhere.

- Discussion Type
- discussion topicworldline formalism
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active 1 day ago

created

*worldline formalism*to go with this Physics.SE answer

- Discussion Type
- discussion topicsolvable group
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Blake Stacey
- Last Active 1 day ago

added some formatting and some cross-links (nilpotent groups!) and added pointer to:

- Cornelia Druţu, Michael Kapovich, Chapter 13 of:
*Geometric group theory*, Colloquium Publications**63**, AMS 2018 (ISBN:978-1-4704-1104-6, pdf)

- Cornelia Druţu, Michael Kapovich, Chapter 13 of:

- Discussion Type
- discussion topicJoseph S. Verducci
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 2 days ago

brief

`category:people`

-entry for hyperlinking references at*Cayley distance*,*Mallows kernel*etc.

- Discussion Type
- discussion topicMichael A. Fligner
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 2 days ago

brief

`category:people`

-entry for hyperlinking references at*Cayley distance*and*Mallows kernel*etc.

- Discussion Type
- discussion topicorbit
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 4
- Last comment by Urs
- Last Active 2 days ago

- Discussion Type
- discussion topiccycle of a permutation
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 2 days ago

- Discussion Type
- discussion topiccycle
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 2 days ago

giving

*cycle of a permutation*its own hyperlink

- Discussion Type
- discussion topicgeometric group theory
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 3 days ago

cross-linked with

*Cayley graph*and added this pointer:- Cornelia Druţu, Michael Kapovich (appendix by Bogdan Nica),
*Geometric group theory*, Colloquium Publications**63**, AMS 2018 (ISBN:978-1-4704-1104-6 pdf)

- Cornelia Druţu, Michael Kapovich (appendix by Bogdan Nica),

- Discussion Type
- discussion topicC. L. Mallows
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 3 days ago

brief

`category:people`

-entry for hyperlinking references at*Mallows kernel*

- Discussion Type
- discussion topicKendall tau distance
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active 3 days ago

starting something, mainly to have a place to record

- Yunlong Jiao, Jean-Philippe Vert,
*The Kendall and Mallows Kernels for Permutations*, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 40, no. 7, pp. 1755-1769, 1 July 2018 (doi:10.1109/TPAMI.2017.2719680, hal:01279273)

- Yunlong Jiao, Jean-Philippe Vert,

- Discussion Type
- discussion topicMaurice G. Kendall
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 3 days ago

brief

`category:people`

-entry for hyperlinking references at*Kendall tau distance*

- Discussion Type
- discussion topicBanach space
- Category Latest Changes
- Started by Mark Meckes
- Comments 31
- Last comment by David_Corfield
- Last Active 3 days ago

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.)

- Discussion Type
- discussion topicJean-Philippe Vert
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 3 days ago

brief

`category:people`

-entry for hyperlinking references at*Mallows kernel*

- Discussion Type
- discussion topicYunlong Jiao
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 3 days ago

brief

`category:people`

-entry for hyperlinking references at*Mallows kernel*

- Discussion Type
- discussion topictransposition permutation
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 3 days ago

- Discussion Type
- discussion topicBanach-Tarski paradox
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active 3 days ago

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

- Discussion Type
- discussion topicdistance (graph theory)
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 3 days ago

some minimum, just for completeness for links at

*Cayley graph*,*finitely generated group*and*geometric group theory*

- Discussion Type
- discussion topicfinitely generated group
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 3 days ago

some minimum, to connect to

*Cayley graph*and*geometric group theory*

- Discussion Type
- discussion topicBogdan Nica
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 3 days ago

brief

`category:people`

-entry for hyperlinking references at*geometric group theory*

- Discussion Type
- discussion topicCornelia Druţu
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 3 days ago

brief

`category:people`

-entry for hyperlinking references at*geometric group theory*

- Discussion Type
- discussion topicDonovan van Osdol
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active 3 days ago

- Discussion Type
- discussion topicHeinrich Weber
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active 3 days ago

- Discussion Type
- discussion topicgroup
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Dmitri Pavlov
- Last Active 3 days ago

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.

- Discussion Type
- discussion topiccomma category
- Category Latest Changes
- Started by Urs
- Comments 21
- Last comment by Richard Williamson
- Last Active 3 days ago

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 $(x,y)$ (or $[x,y]$ as is now more common) for a hom-set. But personally, I like $(f \rightarrow g)$ (or $(f \searrow g)$ if you want to differentiate from a cocomma category, but that seems an unlikely confusion), as it is a category of arrows from $f$ to $g$. —Toby Bartels

Mike: Perhaps. I never write $(x,y)$ for a hom-set, only $A(x,y)$ or $hom_A(x,y)$ where $A$ is the category involved, and this is also the common practice in nearly all mathematics I have read. I have seen $[x,y]$ 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) $E(f,g)$ or $hom_E(f,g)$

*if*you are considering it as a discrete fibration from $A$ to $B$. 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 $(f/g)$ as less visually distracting, and evidently a generalization of the common notation $C/x$ for a slice category.*Toby*: Well, I never stick ‘$E$’ 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 $(f,g)$ for either comma objects or hom-sets is that it’s already such an overloaded notation. My first thought when I see $(f,g)$ in a category is that we have $f:X\to A$ and $g:X\to B$ and we’re talking about the pair $(f,g):X\to A\times B$ — surely also a natural generalization of the*very*well-established notation for ordered pairs.*Toby*: The notation $(f/g/h)$ for a double comma object makes me like $(f \to g \to h)$ even more!*Mike*: I’d rather avoid using $\to$ in the name of an object; talking about projections $p:(f\to g)\to A$ looks a good deal more confusing to me than $p:(f/g)\to A$.*Toby*: I can handle that, but after thinking about it more, I've realised that the arrow doesn't really work. If $f, g: A \to B$, then $f \to g$ ought to be the set of transformations between them. (Or $f \Rightarrow g$, 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 $(f,g)$ 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 $(f,g)$ 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 $f$ and $g$ 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 $C$ to $D$, that are hom-sets. Finally, I don’t think the notation $(f,g)$ scales well to double comma objects; we could write $(f,g,h)$ 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 $M[C\stackrel{f}{\to} E \stackrel{g}{\leftarrow} D]$. Maybe $comma[C\stackrel{f}{\to} E \stackrel{g}{\leftarrow} D]$? Lengthy, but at least unambiguous. Or maybe ${}_f {E^I}_g$?

Zoran Skoda: $(f/g)$ or $(f\downarrow g)$ are the only two standard notations nowdays, I think the original $(f,g)$ which was done for typographical reasons in archaic period is abandonded by the LaTeX era. $(f/g)$ is more popular among practical mathematicians, and special cases, like when $g = id_D$) and $(f\downarrow g)$ among category experts…other possibilities for notation should be avoided I think.

Urs: sounds good. I’ll try to stick to $(f/g)$ then.

Mike: There are many category theorists who write $(f/g)$, including (in my experience) most Australians. I prefer $(f/g)$ myself, although I occasionally write $(f\downarrow g)$ if I’m talking to someone who I worry might be confused by $(f/g)$.

Urs: recently in a talk when an over-category appeared as $C/a$ somebody in the audience asked: “What’s that quotient?”. But $(C/a)$ already looks different. And of course the proper $(Id_C/const_a)$ even more so.

Anyway, that just to say: i like $(f/g)$, find it less cumbersome than $(f\downarrow g)$ and apologize for having written $(f,g)$ so often.

*Toby*: I find $(f \downarrow g)$ more self explanatory, but $(f/g)$ is cool. $(f,g)$ was reasonable, but we now have better options.=–

- Discussion Type
- discussion topicChristopher D. Hollings
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active 3 days ago

- Discussion Type
- discussion topicAnton Suschkewitsch
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 3
- Last comment by Dmitri Pavlov
- Last Active 3 days ago

Created:

Anton Suschkewitsch (Антон Казимирович Сушкевич) was a Russian mathematician working on semigroup theory.

## Selected writings

- Anton Suschkewitsch,
*On a generalization of the associative law*. Transactions of the American Mathematical Society 31:1 (1929), 204–204. doi.

- Anton Suschkewitsch,

- Discussion Type
- discussion topicheap
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 7
- Last comment by Dmitri Pavlov
- Last Active 3 days ago

- Discussion Type
- discussion topictorsor
- Category Latest Changes
- Started by nLab edit announcer
- Comments 40
- Last comment by zskoda
- Last Active 3 days ago

- Discussion Type
- discussion topiccommutative square
- Category Latest Changes
- Started by varkor
- Comments 1
- Last comment by varkor
- Last Active 3 days ago

- Discussion Type
- discussion topicextranatural transformation
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active 3 days ago

- Discussion Type
- discussion topiccycle category
- Category Latest Changes
- Started by Todd_Trimble
- Comments 9
- Last comment by Tim Campion
- Last Active 3 days ago

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 $\mathbb{Z}$.

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 $[n]_\Lambda$ which are freely generated by the graph $0\to 1\to 2\to\ldots\to n\to 0$, and whose morphisms $\Lambda([m],[n])\subset\mathrm{Cat}([m],[n])$ are precisely the functors of degree $1$ (seen either at the level of nerves or via the embedding $\mathrm{Ob}[n]_\Lambda\to \mathbf{R}/\mathbf{Z}\cong S^1$ given by $k\mapsto k/(n+1)\,\mathrm{mod}\,\mathbf{Z}$ on the level of objects, the rest being obvious).

The simplex category $\Delta$ can be identified with a subcategory of $\Lambda$, having the same objects but with fewer morphisms. This identification does not respect the inclusions into $Cat$, however, since $[n]$ and $[n]_\Lambda$ are different categories.

- Discussion Type
- discussion topiccompactly generated topological space
- Category Latest Changes
- Started by Todd_Trimble
- Comments 37
- Last comment by martinescardo
- Last Active 3 days ago

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.

- Discussion Type
- discussion topicbackpropagation
- Category Latest Changes
- Started by Matthijs Vákár
- Comments 2
- Last comment by zskoda
- Last Active 4 days ago

- Discussion Type
- discussion topicempty function
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Mike Shulman
- Last Active 4 days ago

- Discussion Type
- discussion topicdifferential programming
- Category Latest Changes
- Started by Matthijs Vákár
- Comments 1
- Last comment by Matthijs Vákár
- Last Active 4 days ago

- Discussion Type
- discussion topicMichèle Raynaud
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 4 days ago

- Discussion Type
- discussion topicpseudo-torsor
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active 4 days ago

for completeness, with pointer to

- Alexander Grothendieck et al., 16.5.15 in:
*Éléments de géométrie algébrique*IV_4. Étude locale des schémas et des morphismes de schémas (Quatrième partie) Inst. Hautes Études Sci. Publ. Math. 32 (1967), 5–361. Ch.IV.§16–21 (numdam:PMIHES_1967__32__5_0)

- Alexander Grothendieck et al., 16.5.15 in: