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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.
added cross-link with compact closed dagger category
tried to add cross-links with the recent slew of related articles
but may have missed some
There should be an explanation on this article, as well as in the identity-on-objects functor article and the principle of equivalence article, why the naive definitions of such an identity-on-objects functor in type theory, using paths/identifications or equivalences, violates the principle of equivalence, and why it is necessary to stipulate that the two object types of the categories are judgmentally equal to each other.
Remark 2.1. In certain parts of the type theory literature, these are also called dagger precategories. However, in the nLab we adopt the view that the objects of a dagger category form a type or infinity-groupoid, in the same way that the objects of a category form a type.
Do we? The nlab is certainly not systematically written in a way that types/infinity-groupoids are the basic notion.
@maxsnew
The debate in type theory is whether the term “category” should refer to precategory or univalent category, since the structure where the type of objects form a set is already called a strict category. By definitions elsewhere in the nLab, where categories come with a class of objects, and don’t have the Rezk completion condition that result in univalent categories, the parallel here would be the precategory definition, where categories come with a type of objects and similarly don’t have the Rezk completion condition.
However, I agree that referring to infinity-groupoids is hugely misleading, since types are only infinity-groupoids if the type theory additionally has the univalence axiom and all higher inductive types. There are type theories where instead of univalence, there is some axiom that contradicts univalence, such as Thomas Streicher’s groupoid model of types, where every type is 1-truncated and thus only a groupoid.
get rid of reference to infinity-groupoid in remark: not all type theories model infinity-groupoids
Anonymous
@22 Just to note, the article on the principle of equivalence has the following sentence:
Philosophically, the concept of strict category is not in itself a breaking of equivalence-invariance; what does break equivalence-invariance is to say that $Set$ (and other well-known categories such as $Grp$, etc) are strict categories.
A structural set theory can certainly say what it means for a functor to be literally the identity on objects (not just a bijection), as long as the two categories involved have (definitionally) the same set of objects, which is the case for a category and its opposite. And I believe this is important to do in giving a correct definition of dagger-category. The same is true in extensional type theory.
Also, I don’t think we can literally used displayed categories to define dagger-categories. I think #21 was just making an analogy.
A number of changes:
Add reference to symmetric bicategory.
Mention terminology “symmetric category”. To be honest, I find this terminology more evocative and consistent than “dagger category” (which I see as a concept with an attitude), but the dagger category terminology has clearly won out for now.
Simon Henry came up with 3 definitions of an $(\infty,1)$-dagger category in response to a MathOverflow question I asked last year: https://mathoverflow.net/questions/427302/n-1-dagger-categories/427322#427322
I could either add the definitions to the section titled “$(\infty,1)$-$\dagger$-categories” on this page, or I could create a separate page on these structures. Which do you think is the better choice?
Is it helpful to give tentative definitions when they have not been used in any work yet? Perhaps it would be better simply to include a link to the MathOverflow discussion in the references?
varkor,
from what I recall, the section on $\dagger$-simplicial sets seems to have only appeared in the CategoryTheory mailing list on Jan 6, 2010, so I’m not sure if inserting Henry’s definitions from the MathOverflow thread into the $(\infinity,1)$-dagger category section is any different from Urs Schreiber inserting Joyal’s definition of $\dagger$-simplicial sets from the mailing list in revision 12 of this article.
video recording of Johnson-Freyd’s talk now here: YT
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