# Start a new discussion

## Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

## Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• starting something – not done yet

• Starting a stub.

• Page created, but author did not leave any comments.

• Created with the following content:

## 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 finite sum

$\sum_{i\in I} m_i\otimes f_i.$

$\sum_{i\in I} m_i f_i(p) = p,\qquad p\in M$ $\sum_{i\in I} r(m_i) f_i = r, \qquad r\in Hom_R(M,R).$

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

• fixed spelling of the word constraint (one instance) in paragraph under ### Gravity as a (non-)gauge theory

cofo

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

• brief category:people-entry for hyperlinking references at M5-brane

• 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

• A stub for now.

• Created page, mainly to record a bunch of references that I am trying to collect. Additional suggested references would be welcome!

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

but I suppose now it must be so

• 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.

• Added an alternative description of the category of orbits via normalizers.

• a minimum, just for completeness

• brief category:people-entry for hyperlinking references at Mallows kernel

• 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

• for completeness, to be able to speak of the generators of the symmetric group

• 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

• Page created, but author did not leave any comments.

• 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 $(x,y)$ (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.

=–

• Page created, but author did not leave any comments.

• 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.

• fix wrong definition of free group action

Alexey Muranov

• Made the double category structure more explicit.

• References for pseudoextranatural transformations by Vidal and Tur 2010 and Corner 2019.

• 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.

• 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.

• Page created, but author did not leave any comments.

• added mentioning of the generalization of “empty function” to “empty morphism” as any morphism out of a strict initial object

• Page created, but author did not leave any comments.

• brief category:people-entry for hyperlinking references

• 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)