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

]]>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.

]]>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:

$(\text{-}) \otimes A \;\; \dashv \;\; (\text{-}) \otimes A^\ast \;\; \dashv \;\; (\text{-}) \otimes A \,.$ ]]>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

]]>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”)

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

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

]]>added pointer to these lecture notes:

- Dominic Culver, Mitchell Faulk,
*Duality Notes*, lecture notes for*West Coast Algebraic Topology Summer School*(2014) [pdf, mathtube]

Added an early reference.

]]>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)

]]>added (here) the example of adjoint endofunctors

(moved from *rigid monoidal category* following discussion there)

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]

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]

Added the example of dualizable modules.

]]>Deleted duplicate redirect.

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

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

]]>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. :-)

]]>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…

]]>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).)

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.

]]>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.

]]>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.

]]>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”.

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