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edited dualizable object a little, added a brief paragraph on dualizable objects in symmetric monoidal $(\infty,n)$-categories
Added “Idea” and “Examples” sections to dualizable object.
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 $X$…
Anything useful in this, from bottom of p. 4?
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?)
Ok, added to references.
Thanks. That’s a nice reference. I have expanded a bit more at dual object.
I went ahead and actually wrote out the explicit definition at dualizable objects (in other words: I typeset the full triangle identities).
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”.
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.
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.
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.
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).)
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 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. :-)
I added to dualizable object a section about duals in linearly distributive categories.
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:
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 these lecture notes:
added pointer to
which has early discussion of dualizable objects (calling them “finite objects”)
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
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:
$(\text{-}) \otimes A \;\; \dashv \;\; (\text{-}) \otimes A^\ast \;\; \dashv \;\; (\text{-}) \otimes A \,.$on top of the list of “original articles” on dualizable objects I have added pointer to
who take aspects of the the strong duality of finite-dimensional vector spaces as the very motivation of category theory.
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