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I took the liberty of incorporating material from Andre Joyal's latest message to the CatTheory mailing list into the entry dagger-category:
created sections
Joyal really made a surprising, subtle and beautiful point. So far the most interesting remark in the whole dagger discussion, to my taste.
Does anyone consider a bicategory of dagger-catgeories? Or dagger-functor categories? Hence a closed monoidal structure on dagger categories? Things like that?
I think the category of dagger-categories and dagger-functors is actually cartesian closed. Giving you in particular a "locally-dagger" 2-category of dagger categories.
What's the dagger structure on the internal hom?
Oh, right, sorry, I get it.
started The category fo dagger-categories
but have to interrup now-- have to dash to get some dinner...
Yes, best typo of all time.
Please do not fix it.
The Category fo Dagger Categories, the place where all the cool cats like to go.
Yes, best typo of all time.
Thanks, I am practicing typos a lot.
anyway: dinner is over, here are more details: the category of dagger-categories
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<blockquote><br/>The Category fo Dagger Categories, the place where all the cool cats like to go.<br/></blockquote><br/>Yo.
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I’m interested in learning more about the contents of Joyal’s email to the categories mailing list from Jan 6, 2010 about dagger categories, but I wasn’t subscribed to the list at the time and I haven’t figured out how to access the archives at gmane. Could somebody post a copy of Joyal’s email? I’m particularly interested to know his definition of infinity dagger category and whether he constructed a full model structure for them.
There’s a copy of the archives here (by Simon Burton). Joyal’s main message is 5477 (Jan 5) with 5487 (Jan 6) as a follow-up.
Thanks so much! Apparently Joyal didn’t really write anything more than is found on the nlab page.
Curious: Joyal’s site of choice for his $\dagger$ simplicial sets – the site of nonempty finite ordinals and maps which preserve or reverse order – is not a full subcategory of the category of dagger categories. For instance, $\Delta[1]$ is the walking arrow, but the free dagger category on an arrow has infinitely many morphisms, so that $Hom(\Delta[1],\Delta[1])$ is not correct for $\Delta[1]$ to be identified with this dagger category. Relatedly, in Joyal’s putative model structure, the representables won’t be fibrant.
I think the site is an Eilenberg-Zilber category, but it has nontrivial automorphisms. Relatedly, the obvious interval object given by the walking unitary isomorphism is not cofibrant. So there seem to be technical difficulties, at any rate, in constructing this model structure.
By the way, if there are any links or quotations still broken or missing in the entry, then – as a service to the community and to the next reader who may have the same question as in #11 – please fix or else add them.
Fix links to Joyal’s posts, as it seems Gmane is not coming back anytime soon.
For more information on what happened to Gmane, see this blog post by Lars Ingebrigtsen.
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