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nLab > Latest Changes: dualizable object

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeDec 13th 2010
- (edited Dec 5th 2013)
- PermaLink
Author: Urs
Format: MarkdownItexedited [[dualizable object]] a little, added a brief paragraph on dualizable objects in symmetric monoidal $(\infty,n)$-categories

edited dualizable object a little, added a brief paragraph on dualizable objects in symmetric monoidal $(\infty,n)$ -categories
- CommentRowNumber2.
- CommentAuthorMike Shulman
- CommentTimeJun 11th 2011
- PermaLink
Author: Mike Shulman
Format: MarkdownItexAdded "Idea" and "Examples" sections to [[dualizable object]].

Added “Idea” and “Examples” sections to dualizable object.
- CommentRowNumber3.
- CommentAuthorUrs
- CommentTimeDec 5th 2013
- (edited Dec 5th 2013)
- PermaLink
Author: Urs
Format: MarkdownItexI have cross-linked _[[dualizable object]]_ and _[[dual object in a closed category]]. The latter I also linked to from _[[closed category]]_. Added the notation $\mathbb{D}X \coloneqq [X,1]$. There should still be more discussion on sufficient conditions for $\mathbb{D}X$ to be a monoidal dual for $X$...

I have cross-linked dualizable object and dual object in a closed category. The latter I also linked to from _closed category. Added the notation $\mathbb{D}X \coloneqq [X,1]$ .

There should still be more discussion on sufficient conditions for $\mathbb{D}X$ to be a monoidal dual for $X$ …
- CommentRowNumber4.
- CommentAuthorDavid_Corfield
- CommentTimeDec 5th 2013
- PermaLink
Author: David_Corfield
Format: MarkdownItexAnything useful in [this](http://www.math.purdue.edu/~gottlieb/Bibliography/53.pdf), from bottom of p. 4?

Anything useful in this, from bottom of p. 4?
- CommentRowNumber5.
- CommentAuthorUrs
- CommentTimeDec 5th 2013
- PermaLink
Author: Urs
Format: MarkdownItexYes, maybe a pointer to that page 5 would be good at _[[dualizable object]]_. (I have to go offline right now, maybe you could add it?)

Yes, maybe a pointer to that page 5 would be good at dualizable object.

(I have to go offline right now, maybe you could add it?)
- CommentRowNumber6.
- CommentAuthorDavid_Corfield
- CommentTimeDec 5th 2013
- PermaLink
Author: David_Corfield
Format: MarkdownItexOk, added to references.

Ok, added to references.
- CommentRowNumber7.
- CommentAuthorUrs
- CommentTimeDec 6th 2013
- PermaLink
Author: Urs
Format: MarkdownItexThanks. That's a nice reference. I have expanded a bit more at _[[dual object]]_.

Thanks. That’s a nice reference. I have expanded a bit more at dual object.
- CommentRowNumber8.
- CommentAuthorUrs
- CommentTimeNov 16th 2016
- PermaLink
Author: Urs
Format: MarkdownItexI went ahead and actually wrote out the explicit definition at _[[dualizable objects]]_ (in other words: I typeset the full triangle identities).

I went ahead and actually wrote out the explicit definition at dualizable objects (in other words: I typeset the full triangle identities).
- CommentRowNumber9.
- CommentAuthorCharles Rezk
- CommentTimeFeb 25th 2017
- PermaLink
Author: Charles Rezk
Format: MarkdownItexSo [dualizable object](https://ncatlab.org/nlab/show/dualizable+object) offers the following as its definition: *** An object $A$ in a [[monoidal category]] $(\mathcal{C}, \otimes, 1)$ is **dualizable** if it has an [[adjunction|adjoint]] when regarded as a [[morphism]] in the one-object [[delooping]] [[bicategory]] $\mathbf{B}\mathcal{C}$ corresponding to $\mathcal{C}$. Its adjoint in $\mathbf{B}\mathcal{C}$ is called its **dual** in $C$ and often written as $A^*$. *** It then goes on to assert that this is equivalent to a more conventional definition (involving certain maps $A^*\otimes A\to 1$, $1\to A\otimes A^*$, etc.). I can't figure out what the above definition is saying, because the links don't explain to me what a "delooping bicategory" is supposed to be. (They tell me what "delooping" is, and what a "bicategory" is. This does not help.) One guess is that $\mathbf{B}\mathcal{C}$ is supposed to have one object, and the endomorphism category of that object would be the functor category $\mathrm{Fun}(\mathcal{C},\mathcal{C})$ (in which case its actually a 2-category). Thus you would "regard" an object of $\mathcal{C}$ as a morphism in $\mathbf{B}\mathcal{C}$ by means of a functor $\mathcal{C} \to \mathrm{Fun}(\mathcal{C},\mathcal{C})$ sending $X$ to $X\otimes -$. If this is so, then its not clear to me that the asserted equivalence of definitions holds, because this $\mathcal{C} \to \mathrm{Fun}(\mathcal{C},\mathcal{C})$ is not likely to be fully faithful. Perhaps what is meant is that $\mathbf{B}\mathcal{C}$ is built, so that the endomorphisms of the object are a category whose objects are $(F,\gamma)$, where $F\colon \mathcal{C}\to \mathcal{C}$ is a functor, and $\gamma$ is a natural isomorphism $\gamma_{X,Y}\colon F(X)\otimes Y \to F(X\otimes Y)$, satisfying some evident properties ("right-module endofunctors", i.e., the thing used in the usual proof of the MacLane strictness theorem), so that we get a fully faithful embedding. This seems more plausible, but it is certainly not spelled out. Another way to describe the issue is as follows. Given an object $A$, suppose we know that the endofunctor $A\otimes-$ admits a right adjoint, which I'll denote $[A,-]$ (note that I'm not assuming that internal hom exists in general, just for $A$). The first way of reading the above definition appears to assert that $A$ is dualizable if such a right adjoint exists, and if there exists an object $A^*$ and a natural isomorphism $A^*\otimes- \approx [A,-]$ of functors. I don't believe this is correct, because it does not specify enough about the nature of the natural isomorphism. Here's a statement which I believe is correct: $A$ is dualizable iff there is a right adjoint functor $[A,-]$ such that the "evident" natural map $A^*\otimes X \to [A,X]$ is a natural isomorphism, where $A^*:=[A,1]$, the "evident" map being constructed from the data of the adjunction. This is Proposition 2.3 of Deligne, "Categories Tannakiennes".

So dualizable object offers the following as its definition:

An object $A$ in a monoidal category $(\mathcal{C}, \otimes, 1)$ is dualizable if it has an adjoint when regarded as a morphism in the one-object delooping bicategory $\mathbf{B}\mathcal{C}$ corresponding to $\mathcal{C}$ . Its adjoint in $\mathbf{B}\mathcal{C}$ is called its dual in $C$ and often written as $A^*$ .

It then goes on to assert that this is equivalent to a more conventional definition (involving certain maps $A^*\otimes A\to 1$ , $1\to A\otimes A^*$ , etc.). I can’t figure out what the above definition is saying, because the links don’t explain to me what a “delooping bicategory” is supposed to be. (They tell me what “delooping” is, and what a “bicategory” is. This does not help.)

One guess is that $\mathbf{B}\mathcal{C}$ is supposed to have one object, and the endomorphism category of that object would be the functor category $\mathrm{Fun}(\mathcal{C},\mathcal{C})$ (in which case its actually a 2-category). Thus you would “regard” an object of $\mathcal{C}$ as a morphism in $\mathbf{B}\mathcal{C}$ by means of a functor $\mathcal{C} \to \mathrm{Fun}(\mathcal{C},\mathcal{C})$ sending $X$ to $X\otimes -$ . If this is so, then its not clear to me that the asserted equivalence of definitions holds, because this $\mathcal{C} \to \mathrm{Fun}(\mathcal{C},\mathcal{C})$ is not likely to be fully faithful.

Perhaps what is meant is that $\mathbf{B}\mathcal{C}$ is built, so that the endomorphisms of the object are a category whose objects are $(F,\gamma)$ , where $F\colon \mathcal{C}\to \mathcal{C}$ is a functor, and $\gamma$ is a natural isomorphism $\gamma_{X,Y}\colon F(X)\otimes Y \to F(X\otimes Y)$ , satisfying some evident properties (“right-module endofunctors”, i.e., the thing used in the usual proof of the MacLane strictness theorem), so that we get a fully faithful embedding. This seems more plausible, but it is certainly not spelled out.

Another way to describe the issue is as follows. Given an object $A$ , suppose we know that the endofunctor $A\otimes-$ admits a right adjoint, which I’ll denote $[A,-]$ (note that I’m not assuming that internal hom exists in general, just for $A$ ). The first way of reading the above definition appears to assert that $A$ is dualizable if such a right adjoint exists, and if there exists an object $A^*$ and a natural isomorphism $A^*\otimes- \approx [A,-]$ of functors. I don’t believe this is correct, because it does not specify enough about the nature of the natural isomorphism.

Here’s a statement which I believe is correct: $A$ is dualizable iff there is a right adjoint functor $[A,-]$ such that the “evident” natural map $A^*\otimes X \to [A,X]$ is a natural isomorphism, where $A^*:=[A,1]$ , the “evident” map being constructed from the data of the adjunction. This is Proposition 2.3 of Deligne, “Categories Tannakiennes”.
- CommentRowNumber10.
- CommentAuthorTodd_Trimble
- CommentTimeFeb 25th 2017
- PermaLink
Author: Todd_Trimble
Format: MarkdownItexCharles, I believe what is meant is that $\mathbf{B}(\mathcal{C})$ has one object, and the endomorphism category at that object is $\mathcal{C}$, with 1-cells composed by the monoidal product. This delooping is parallel to what we call around here the delooping of a group $G$, namely the one-object category whose hom is given by $G$. I think under this reading, the nLab article is correct.

Charles, I believe what is meant is that $\mathbf{B}(\mathcal{C})$ has one object, and the endomorphism category at that object is $\mathcal{C}$ , with 1-cells composed by the monoidal product. This delooping is parallel to what we call around here the delooping of a group $G$ , namely the one-object category whose hom is given by $G$ . I think under this reading, the nLab article is correct.
- CommentRowNumber11.
- CommentAuthorMike Shulman
- CommentTimeFeb 26th 2017
- PermaLink
Author: Mike Shulman
Format: MarkdownItexYes, that's what is meant. But it's clearly a problem if we use that term and can't give a link that succintly explains what it means! Should have a dedicated page called [[delooping bicategory]]? It's certainly an important idea.

Yes, that’s what is meant. But it’s clearly a problem if we use that term and can’t give a link that succintly explains what it means! Should have a dedicated page called delooping bicategory? It’s certainly an important idea.
- CommentRowNumber12.
- CommentAuthorTodd_Trimble
- CommentTimeFeb 26th 2017
- PermaLink
Author: Todd_Trimble
Format: MarkdownItexWell, delooping can be seen to apply to structures in the periodic table, so that's where I'd put the level of generality. The notion of delooping bicategory could be in a list of examples.

Well, delooping can be seen to apply to structures in the periodic table, so that’s where I’d put the level of generality. The notion of delooping bicategory could be in a list of examples.
- CommentRowNumber13.
- CommentAuthorTodd_Trimble
- CommentTimeFeb 26th 2017
- PermaLink
Author: Todd_Trimble
Format: MarkdownItexOkay, I added a brief section to [[delooping]], titled "Delooping of higher categorical structures". But actually, Charles's question might have been headed off at the pass had "delooping" there pointed instead to [[delooping hypothesis]], which seems to give adequate explanation. (But no need to point it there now, as my edit takes care of it (and also mentions **delooping bicategory** specifically).)

Okay, I added a brief section to delooping, titled “Delooping of higher categorical structures”. But actually, Charles’s question might have been headed off at the pass had “delooping” there pointed instead to delooping hypothesis, which seems to give adequate explanation. (But no need to point it there now, as my edit takes care of it (and also mentions delooping bicategory specifically).)
- CommentRowNumber14.
- CommentAuthorMike Shulman
- CommentTimeFeb 26th 2017
- PermaLink
Author: Mike Shulman
Format: MarkdownItexThat's helpful; but I still think it would be useful to have a page about the delooping bicategory specifically that doesn't require delving into $(n,k+1)$-categories, since it's quite a simple special case and useful to build intuition for the general one. Of course, if I really felt that strongly about it I would write such a page myself...

That’s helpful; but I still think it would be useful to have a page about the delooping bicategory specifically that doesn’t require delving into $(n,k+1)$ -categories, since it’s quite a simple special case and useful to build intuition for the general one.

Of course, if I really felt that strongly about it I would write such a page myself…
- CommentRowNumber15.
- CommentAuthorTodd_Trimble
- CommentTimeFeb 27th 2017
- PermaLink
Author: Todd_Trimble
Format: MarkdownItexOkay, I added some nuts-and-bolts details on the delooping bicategory to [[delooping]]; if anyone is so inclined, one could copy and paste to create a stub for [[delooping bicategory]]. :-)

Okay, I added some nuts-and-bolts details on the delooping bicategory to delooping; if anyone is so inclined, one could copy and paste to create a stub for delooping bicategory. :-)
- CommentRowNumber16.
- CommentAuthorMike Shulman
- CommentTimeOct 20th 2017
- PermaLink
Author: Mike Shulman
Format: MarkdownItexI added to [[dualizable object]] a section about duals in [[linearly distributive categories]].

I added to dualizable object a section about duals in linearly distributive categories.
- CommentRowNumber17.
- CommentAuthorMike Shulman
- CommentTimeMar 12th 2019
- PermaLink
Author: Mike Shulman
Format: MarkdownItexAdded remark about different definitions and non-definitions of dualizability. <a href="https://ncatlab.org/nlab/revision/diff/dualizable+object/39">diff</a>, <a href="https://ncatlab.org/nlab/revision/dualizable+object/39">v39</a>, <a href="https://ncatlab.org/nlab/show/dualizable+object">current</a>

Added remark about different definitions and non-definitions of dualizability.

diff, v39, current
- CommentRowNumber18.
- CommentAuthorDmitri Pavlov
- CommentTimeAug 23rd 2019
- PermaLink
Author: Dmitri Pavlov
Format: MarkdownItexDeleted duplicate redirect. <a href="https://ncatlab.org/nlab/revision/diff/dualizable+object/41">diff</a>, <a href="https://ncatlab.org/nlab/revision/dualizable+object/41">v41</a>, <a href="https://ncatlab.org/nlab/show/dualizable+object">current</a>

Deleted duplicate redirect.

diff, v41, current
- CommentRowNumber19.
- CommentAuthorDmitri Pavlov
- CommentTimeMay 23rd 2022
- PermaLink
Author: Dmitri Pavlov
Format: MarkdownItexAdded the example of dualizable modules. <a href="https://ncatlab.org/nlab/revision/diff/dualizable+object/44">diff</a>, <a href="https://ncatlab.org/nlab/revision/dualizable+object/44">v44</a>, <a href="https://ncatlab.org/nlab/show/dualizable+object">current</a>

Added the example of dualizable modules.

diff, v44, current
- CommentRowNumber20.
- CommentAuthorUrs
- CommentTimeOct 3rd 2022
- PermaLink
Author: Urs
Format: MarkdownItexadded pointer to original articles: * [[Pierre Deligne]], [[James Milne]]: §1 of: *Tannakian categories*, in: *Hodge Cycles, Motives, and Shimura Varieties*, Lecture Notes in Mathematics **900**, Springer (1982) [[doi:10.1007/978-3-540-38955-2_4](https://doi.org/10.1007/978-3-540-38955-2_4), [webpage](http://www.jmilne.org/math/xnotes/tc.html)] * [[Albrecht Dold]], [[Dieter Puppe]], §1 of: *Duality, Trace and Transfer*, Proceedings of the Steklov Institute of Mathematics, **154** (1984) 85–103 [[mathnet:tm2435](http://mi.mathnet.ru/tm2435), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/doldpup2.pdf)] and to the review in: * [[Peter May]], §2 in: *Picard Groups, Grothendieck Rings, and Burnside Rings of Categories*, Advances in Mathematics **163** 1 (2001) 1-16 [[pdf](http://math.uchicago.edu/~may/PAPERS/PicJan01.pdf), [doi:10.1006/aima.2001.1996](https://doi.org/10.1006/aima.2001.1996)] <a href="https://ncatlab.org/nlab/revision/diff/dualizable+object/45">diff</a>, <a href="https://ncatlab.org/nlab/revision/dualizable+object/45">v45</a>, <a href="https://ncatlab.org/nlab/show/dualizable+object">current</a>
added pointer to original articles:
- Pierre Deligne, James Milne: §1 of: Tannakian categories, in: Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Mathematics 900, Springer (1982) [doi:10.1007/978-3-540-38955-2_4, webpage]
- Albrecht Dold, Dieter Puppe, §1 of: Duality, Trace and Transfer, Proceedings of the Steklov Institute of Mathematics, 154 (1984) 85–103 [mathnet:tm2435, pdf]
and to the review in:
- Peter May, §2 in: Picard Groups, Grothendieck Rings, and Burnside Rings of Categories, Advances in Mathematics 163 1 (2001) 1-16 [pdf, doi:10.1006/aima.2001.1996]
diff, v45, current
- CommentRowNumber21.
- CommentAuthorUrs
- CommentTimeOct 3rd 2022
- (edited Oct 3rd 2022)
- PermaLink
Author: Urs
Format: MarkdownItexalso these pointers: * [[Mark Hovey]], [[John Palmieri]], [[Neil Strickland]]: Def. 1.1.2 and Thm. 2.1.3 in: *Axiomatic stable homotopy theory*, Memoirs Amer. Math. Soc. **610** (1997) [[ISBN:978-1-4704-0195-5](https://bookstore.ams.org/memo-128-610), [pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/axiomatic.pdf)] * [[L. Gaunce Lewis]], [[Peter May]], [[Mark Steinberger]] (with contributions by [[James McClure|J.E. McClure]]): §III.1 in: *Equivariant stable homotopy theory*, Springer Lecture Notes in Mathematics **1213** (1986) [[pdf](http://www.math.uchicago.edu/~may/BOOKS/equi.pdf), [doi:10.1007/BFb0075778](https://link.springer.com/book/10.1007/BFb0075778)] <a href="https://ncatlab.org/nlab/revision/diff/dualizable+object/45">diff</a>, <a href="https://ncatlab.org/nlab/revision/dualizable+object/45">v45</a>, <a href="https://ncatlab.org/nlab/show/dualizable+object">current</a>
also these pointers:
- Mark Hovey, John Palmieri, Neil Strickland: Def. 1.1.2 and Thm. 2.1.3 in: Axiomatic stable homotopy theory, Memoirs Amer. Math. Soc. 610 (1997) [ISBN:978-1-4704-0195-5, pdf]
- L. Gaunce Lewis, Peter May, Mark Steinberger (with contributions by J.E. McClure): §III.1 in: Equivariant stable homotopy theory, Springer Lecture Notes in Mathematics 1213 (1986) [pdf, doi:10.1007/BFb0075778]
diff, v45, current
- CommentRowNumber22.
- CommentAuthorUrs
- CommentTimeMay 7th 2023
- PermaLink
Author: Urs
Format: MarkdownItexadded ([here](https://ncatlab.org/nlab/show/dualizable+object#AdjointEndofunctors)) the example of adjoint endofunctors (moved from *[[rigid monoidal category]]* following discussion [there](https://nforum.ncatlab.org/discussion/13610/rigid-monoidal-category/?Focus=109304#Comment_109304)) <a href="https://ncatlab.org/nlab/revision/diff/dualizable+object/47">diff</a>, <a href="https://ncatlab.org/nlab/revision/dualizable+object/47">v47</a>, <a href="https://ncatlab.org/nlab/show/dualizable+object">current</a>

added (here) the example of adjoint endofunctors

(moved from rigid monoidal category following discussion there)

diff, v47, current
- CommentRowNumber23.
- CommentAuthorUrs
- CommentTimeMay 7th 2023
- PermaLink
Author: Urs
Format: MarkdownItexTouched the formatting of the whole list of examples [here](https://ncatlab.org/nlab/show/dualizable+object#Examples) (and edited the floating TOCs included in the entry to work around the bug which prevented this link from working) <a href="https://ncatlab.org/nlab/revision/diff/dualizable+object/47">diff</a>, <a href="https://ncatlab.org/nlab/revision/dualizable+object/47">v47</a>, <a href="https://ncatlab.org/nlab/show/dualizable+object">current</a>

Touched the formatting of the whole list of examples here

(and edited the floating TOCs included in the entry to work around the bug which prevented this link from working)

diff, v47, current
- CommentRowNumber24.
- CommentAuthorvarkor
- CommentTimeJun 20th 2023
- PermaLink
Author: varkor
Format: MarkdownItexAdded an early reference. <a href="https://ncatlab.org/nlab/revision/diff/dualizable+object/49">diff</a>, <a href="https://ncatlab.org/nlab/revision/dualizable+object/49">v49</a>, <a href="https://ncatlab.org/nlab/show/dualizable+object">current</a>

Added an early reference.

diff, v49, current
- CommentRowNumber25.
- CommentAuthorUrs
- CommentTimeSep 21st 2023
- PermaLink
Author: Urs
Format: MarkdownItexadded pointer to these lecture notes: * [[Dominic Culver]], [[Mitchell Faulk]], *Duality Notes*, lecture notes for *West Coast Algebraic Topology Summer School* (2014) [[pdf](https://mathtube.org/sites/default/files/lecture-notes/Duality%20Notes.pdf), [mathtube](https://www.mathtube.org/lecture/notes/duality-notes)] <a href="https://ncatlab.org/nlab/revision/diff/dualizable+object/50">diff</a>, <a href="https://ncatlab.org/nlab/revision/dualizable+object/50">v50</a>, <a href="https://ncatlab.org/nlab/show/dualizable+object">current</a>
added pointer to these lecture notes:
- Dominic Culver, Mitchell Faulk, Duality Notes, lecture notes for West Coast Algebraic Topology Summer School (2014) [pdf, mathtube]
diff, v50, current
- CommentRowNumber26.
- CommentAuthorUrs
- CommentTimeSep 21st 2023
- (edited Sep 21st 2023)
- PermaLink
Author: Urs
Format: MarkdownItexWhere it said that $A \otimes (-)$ having a right adjoint does "not seem to" be sufficient for $A$ to be dualizable, I have added pointer to a counterexample, now in the remark [here](https://ncatlab.org/nlab/show/dualizable+object#TensorAdjunctabilityDoesNotImplyDualizability) <a href="https://ncatlab.org/nlab/revision/diff/dualizable+object/50">diff</a>, <a href="https://ncatlab.org/nlab/revision/dualizable+object/50">v50</a>, <a href="https://ncatlab.org/nlab/show/dualizable+object">current</a>

Where it said that $A \otimes (-)$ having a right adjoint does “not seem to” be sufficient for $A$ to be dualizable,

I have added pointer to a counterexample, now in the remark here

diff, v50, current
- CommentRowNumber27.
- CommentAuthorUrs
- CommentTimeSep 21st 2023
- (edited Sep 21st 2023)
- PermaLink
Author: Urs
Format: MarkdownItexhave made the definition of *reflexive* dualizable objects more explicit ([here](https://ncatlab.org/nlab/show/dualizable+object#ReflexiveDualizableObject)) <a href="https://ncatlab.org/nlab/revision/diff/dualizable+object/50">diff</a>, <a href="https://ncatlab.org/nlab/revision/dualizable+object/50">v50</a>, <a href="https://ncatlab.org/nlab/show/dualizable+object">current</a>

have made the definition of reflexive dualizable objects more explicit (here)

diff, v50, current
- CommentRowNumber28.
- CommentAuthorUrs
- CommentTimeSep 21st 2023
- PermaLink
Author: Urs
Format: MarkdownItexadded pointer to * [[Bodo Pareigis]], *Non-additive ring and module theory IV: The Brauer group of a symmetric monoidal category*, Lecture Notes in Mathematics **549** (1976) [[doi:10.1007/BFb0077339]( https://doi.org/10.1007/BFb0077339)] which has early discussion of dualizable objects (calling them "finite objects") <a href="https://ncatlab.org/nlab/revision/diff/dualizable+object/50">diff</a>, <a href="https://ncatlab.org/nlab/revision/dualizable+object/50">v50</a>, <a href="https://ncatlab.org/nlab/show/dualizable+object">current</a>
added pointer to
- Bodo Pareigis, Non-additive ring and module theory IV: The Brauer group of a symmetric monoidal category, Lecture Notes in Mathematics 549 (1976) [doi:10.1007/BFb0077339]
which has early discussion of dualizable objects (calling them “finite objects”)

diff, v50, current
- CommentRowNumber29.
- CommentAuthorUrs
- CommentTimeSep 21st 2023
- PermaLink
Author: Urs
Format: MarkdownItexand to this thesis * Thomas S. Ligon, *Galois-Theorie in monoidalen Kategorien*, Munich (1978) [[pdf](https://edoc.ub.uni-muenchen.de/14958/1/Ligon_Thomas.pdf), [doi:10.5282/edoc.14958](https://doi.org/10.5282/edoc.14958)] transl: *Galois theory in monoidal categories* (2019) [[pdf](https://edoc.ub.uni-muenchen.de/24952/7/Ligon_Thomas.pdf), [doi:10.5282/edoc.24952](https://doi.org/10.5282/edoc.24952)] following Pareigis <a href="https://ncatlab.org/nlab/revision/diff/dualizable+object/50">diff</a>, <a href="https://ncatlab.org/nlab/revision/dualizable+object/50">v50</a>, <a href="https://ncatlab.org/nlab/show/dualizable+object">current</a>
and to this thesis
- Thomas S. Ligon, Galois-Theorie in monoidalen Kategorien, Munich (1978) [pdf, doi:10.5282/edoc.14958]
  
  transl: Galois theory in monoidal categories (2019) [pdf, doi:10.5282/edoc.24952]
following Pareigis

diff, v50, current
- CommentRowNumber30.
- CommentAuthorUrs
- CommentTimeSep 21st 2023
- (edited Sep 21st 2023)
- PermaLink
Author: Urs
Format: MarkdownItexmade explicit ([here](https://ncatlab.org/nlab/show/dualizable+object#RelationToAdjunctableTensorProducts)) the equivalence between dual objects and their adjunctable tensor product functors *for specific hom-isomorphism* ( [Dold & Puppe 1984, Thm. 1.3 (b) and (c)](https://ncatlab.org/nlab/show/dualizable+object#DoldPuppe84)) and mentioned that the resulting adjunctions are [[amazing right adjoint|amazing]] [[ambidextrous adjunction|ambidextrous]]: $$ (\text{-}) \otimes A \;\; \dashv \;\; (\text{-}) \otimes A^\ast \;\; \dashv \;\; (\text{-}) \otimes A \,. $$ <a href="https://ncatlab.org/nlab/revision/diff/dualizable+object/51">diff</a>, <a href="https://ncatlab.org/nlab/revision/dualizable+object/51">v51</a>, <a href="https://ncatlab.org/nlab/show/dualizable+object">current</a>

made explicit (here) the equivalence between dual objects and their adjunctable tensor product functors for specific hom-isomorphism ( Dold & Puppe 1984, Thm. 1.3 (b) and (c))

and mentioned that the resulting adjunctions are amazing ambidextrous:
$(\text{-}) \otimes A \;\; \dashv \;\; (\text{-}) \otimes A^\ast \;\; \dashv \;\; (\text{-}) \otimes A \,.$
diff, v51, current
- CommentRowNumber31.
- CommentAuthorUrs
- CommentTimeSep 27th 2023
- PermaLink
Author: Urs
Format: MarkdownItexon top of the list of "original articles" on dualizable objects I have added pointer to * [[Samuel Eilenberg]], [[Saunders MacLane]], first page of: *[[General Theory of Natural Equivalences]]*, Transactions of the American Mathematical Society **58** 2 (1945) 231-294 [[doi:10.1090/S0002-9947-1945-0013131-6](https://doi.org/10.1090/S0002-9947-1945-0013131-6), [jstor:1990284](http://www.jstor.org/stable/1990284)] who take aspects of the the strong duality of finite-dimensional vector spaces as the very motivation of category theory. <a href="https://ncatlab.org/nlab/revision/diff/dualizable+object/53">diff</a>, <a href="https://ncatlab.org/nlab/revision/dualizable+object/53">v53</a>, <a href="https://ncatlab.org/nlab/show/dualizable+object">current</a>
on top of the list of “original articles” on dualizable objects I have added pointer to
- Samuel Eilenberg, Saunders MacLane, first page of: General Theory of Natural Equivalences, Transactions of the American Mathematical Society 58 2 (1945) 231-294 [doi:10.1090/S0002-9947-1945-0013131-6, jstor:1990284]
who take aspects of the the strong duality of finite-dimensional vector spaces as the very motivation of category theory.

diff, v53, current
- CommentRowNumber32.
- CommentAuthorUrs
- CommentTimeOct 28th 2023
- PermaLink
Author: Urs
Format: MarkdownItexadded mentioning of the example of dualizable chain complexes ([here](https://ncatlab.org/nlab/show/dualizable+object#DualizableChainComplexes)) <a href="https://ncatlab.org/nlab/revision/diff/dualizable+object/54">diff</a>, <a href="https://ncatlab.org/nlab/revision/dualizable+object/54">v54</a>, <a href="https://ncatlab.org/nlab/show/dualizable+object">current</a>

added mentioning of the example of dualizable chain complexes (here)

diff, v54, current
- CommentRowNumber33.
- CommentAuthorvarkor
- CommentTimeFeb 6th 2024
- PermaLink
Author: varkor
Format: MarkdownItexAdded a reference giving an abstract understanding of the connection between finiteness and dualisability. <a href="https://ncatlab.org/nlab/revision/diff/dualizable+object/57">diff</a>, <a href="https://ncatlab.org/nlab/revision/dualizable+object/57">v57</a>, <a href="https://ncatlab.org/nlab/show/dualizable+object">current</a>

Added a reference giving an abstract understanding of the connection between finiteness and dualisability.

diff, v57, current
- CommentRowNumber34.
- CommentAuthorUrs
- CommentTimeFeb 22nd 2024
- (edited Feb 22nd 2024)
- PermaLink
Author: Urs
Format: MarkdownItexFixed the wording in the remark on adjointability of tensoring with an object not implying the dualizability of that object ([here](https://ncatlab.org/nlab/show/dualizable+object#TensorAdjunctabilityDoesNotImplyDualizability)). <a href="https://ncatlab.org/nlab/revision/diff/dualizable+object/59">diff</a>, <a href="https://ncatlab.org/nlab/revision/dualizable+object/59">v59</a>, <a href="https://ncatlab.org/nlab/show/dualizable+object">current</a>

Fixed the wording in the remark on adjointability of tensoring with an object not implying the dualizability of that object (here).

diff, v59, current
- CommentRowNumber35.
- CommentAuthorJohn Baez
- CommentTimeAug 15th 2024
- PermaLink
Author: John Baez
Format: MarkdownItexAdded link to n-Cafe discussion on subtle problem involving compact closure. <a href="https://ncatlab.org/nlab/revision/diff/dualizable+object/61">diff</a>, <a href="https://ncatlab.org/nlab/revision/dualizable+object/61">v61</a>, <a href="https://ncatlab.org/nlab/show/dualizable+object">current</a>

Added link to n-Cafe discussion on subtle problem involving compact closure.

diff, v61, current

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