Fixed the wording in the remark on adjointability of tensoring with an object not implying the dualizability of that object (here).
]]>Added a reference giving an abstract understanding of the connection between finiteness and dualisability.
]]>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
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:
]]>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
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 having a right adjoint does “not seem to” be sufficient for to be dualizable,
I have added pointer to a counterexample, now in the remark here
]]>added pointer to these lecture notes:
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:
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 -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.
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