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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeFeb 27th 2014
• (edited Feb 27th 2014)

All our entries which wanted to point to something like dualizing module (such as at Verdier duality) or the more general concept in a closed monoidal category (such as at star-autonomous category) used to point to the entry dualizable object, which however did not really discuss this specific concept of “dualizable object”.

Therefore I have now created dualizing object in a closed category and made these entries point to that, instead.

Mentioned that in homological algebra/stable homotopy theory one usually puts additional finiteness conditions on the would-be dualizing object and added a brief remark on Anderson duality as a fundamental example.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMar 5th 2014

briefly added the statement of “Joyal’s lemma” (a cartesian closed category with a dualizing object is a preorder).

• CommentRowNumber3.
• CommentAuthorTodd_Trimble
• CommentTimeMar 5th 2014

I generalized Joyal’s lemma a bit and added a proof.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeMar 5th 2014

Thanks!

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeMar 5th 2014

Is a self-dual Heyting algebra necessarily a Boolean algebra?

• CommentRowNumber6.
• CommentAuthorTodd_Trimble
• CommentTimeMar 5th 2014

Mike, no: the unit interval is a Heyting algebra and has an obvious order-reversing involution.

• CommentRowNumber7.
• CommentAuthorMike Shulman
• CommentTimeMar 6th 2014

Maybe it would be good to mention, then, that when the self-duality comes from a dualizing object we get the stronger conclusion that the Heyting algebra is a Boolean algebra. Unless I’m wrongly remembering that.

• CommentRowNumber8.
• CommentAuthorTodd_Trimble
• CommentTimeMar 6th 2014

No, you’re remembering correctly; I’ll add it in.

• CommentRowNumber9.
• CommentAuthorMike Shulman
• CommentTimeOct 30th 2017

The page dualizing object in a closed category appears to make the claim that the double-dualization map is an isomorphism for all spectra $A$, which is false; this is only true if $A$ is sufficiently finite. One might be able to guess that this is not what is meant by reading sufficiently closely, since the third paragraph of the Idea section of the latter says that “in homological algebra and stable homotopy theory there are typically also certain finiteness conditions imposed”, but this should be made much more explicit.

I would fix it, but I’m not sure what the right fix is. Should “a dualizing object in a closed category” be defined by default to mean one for which double-dualization is always an isomorphism? The page star-autonomous category links to it with that meaning assumed. Or should it be a faithful generalization of the apparently-standard notion of “dualiziing complex”/”dualizing module” in homological algebra and stable homotopy theory, which only implies this property for suitably finite $A$? The page Anderson duality links to it with this meaning assumed. Or should the page dualizing object in a closed category try to unify the two, e.g. by defining a general notion of “dualizing object with respect to a subcategory”?

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeOct 30th 2017

Thanks for the alert. I have tweaked the wording a bit, along the lines of the third option (“unify”). But please edit/expand further as need be.

• CommentRowNumber11.
• CommentAuthorMike Shulman
• CommentTimeOct 31st 2017

Okay, thanks. Unify was my default assumption too; I’ve tried to clean it up some more.