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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeDec 13th 2010
    • (edited Dec 5th 2013)

    edited dualizable object a little, added a brief paragraph on dualizable objects in symmetric monoidal (,n)(\infty,n)-categories

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeJun 11th 2011

    Added “Idea” and “Examples” sections to dualizable object.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 5th 2013
    • (edited Dec 5th 2013)

    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 𝔻X[X,1]\mathbb{D}X \coloneqq [X,1].

    There should still be more discussion on sufficient conditions for 𝔻X\mathbb{D}X to be a monoidal dual for XX

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 5th 2013

    Anything useful in this, from bottom of p. 4?

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeDec 5th 2013

    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

    Ok, added to references.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeDec 6th 2013

    Thanks. That’s a nice reference. I have expanded a bit more at dual object.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeNov 16th 2016

    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

    So dualizable object offers the following as its definition:

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

    It then goes on to assert that this is equivalent to a more conventional definition (involving certain maps A *A1A^*\otimes A\to 1, 1AA *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 B𝒞\mathbf{B}\mathcal{C} is supposed to have one object, and the endomorphism category of that object would be the functor category Fun(𝒞,𝒞)\mathrm{Fun}(\mathcal{C},\mathcal{C}) (in which case its actually a 2-category). Thus you would “regard” an object of 𝒞\mathcal{C} as a morphism in B𝒞\mathbf{B}\mathcal{C} by means of a functor 𝒞Fun(𝒞,𝒞)\mathcal{C} \to \mathrm{Fun}(\mathcal{C},\mathcal{C}) sending XX to XX\otimes -. If this is so, then its not clear to me that the asserted equivalence of definitions holds, because this 𝒞Fun(𝒞,𝒞)\mathcal{C} \to \mathrm{Fun}(\mathcal{C},\mathcal{C}) is not likely to be fully faithful.

    Perhaps what is meant is that B𝒞\mathbf{B}\mathcal{C} is built, so that the endomorphisms of the object are a category whose objects are (F,γ)(F,\gamma), where F:𝒞𝒞F\colon \mathcal{C}\to \mathcal{C} is a functor, and γ\gamma is a natural isomorphism γ X,Y:F(X)YF(XY)\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 AA, suppose we know that the endofunctor AA\otimes- admits a right adjoint, which I’ll denote [A,][A,-] (note that I’m not assuming that internal hom exists in general, just for AA). The first way of reading the above definition appears to assert that AA is dualizable if such a right adjoint exists, and if there exists an object A *A^* and a natural isomorphism A *[A,]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: AA is dualizable iff there is a right adjoint functor [A,][A,-] such that the “evident” natural map A *X[A,X]A^*\otimes X \to [A,X] is a natural isomorphism, where A *:=[A,1]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

    Charles, I believe what is meant is that B(𝒞)\mathbf{B}(\mathcal{C}) has one object, and the endomorphism category at that object is 𝒞\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 GG, namely the one-object category whose hom is given by GG. I think under this reading, the nLab article is correct.

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 26th 2017

    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

    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

    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

    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)(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 26th 2017

    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

    I added to dualizable object a section about duals in linearly distributive categories.

    • CommentRowNumber17.
    • CommentAuthorMike Shulman
    • CommentTimeMar 12th 2019

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

    diff, v39, current

    • CommentRowNumber18.
    • CommentAuthorDmitri Pavlov
    • CommentTimeAug 23rd 2019

    Deleted duplicate redirect.

    diff, v41, current

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