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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJan 6th 2010
• (edited Jan 6th 2010)

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

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeJan 6th 2010

Joyal really made a surprising, subtle and beautiful point. So far the most interesting remark in the whole dagger discussion, to my taste.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeMar 22nd 2010

Does anyone consider a bicategory of dagger-catgeories? Or dagger-functor categories? Hence a closed monoidal structure on dagger categories? Things like that?

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeMar 22nd 2010

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.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeMar 22nd 2010
• (edited Mar 22nd 2010)

What's the dagger structure on the internal hom?

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeMar 22nd 2010

Oh, right, sorry, I get it.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeMar 22nd 2010

but have to interrup now-- have to dash to get some dinner...

• CommentRowNumber8.
• CommentAuthorHarry Gindi
• CommentTimeMar 22nd 2010
• (edited Mar 22nd 2010)

Yes, best typo of all time.

The Category fo Dagger Categories, the place where all the cool cats like to go.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeMar 22nd 2010
• (edited Mar 22nd 2010)

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

• CommentRowNumber10.
• CommentAuthorIan_Durham
• CommentTimeMar 23rd 2010
This comment is invalid XHTML+MathML+SVG; displaying source. <div> <blockquote><br/>The Category fo Dagger Categories, the place where all the cool cats like to go.<br/></blockquote><br/>Yo. </div>
• CommentRowNumber11.
• CommentAuthorTim Campion
• CommentTimeSep 26th 2020

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.

• CommentRowNumber12.
• CommentAuthorUlrik
• CommentTimeSep 27th 2020
• (edited Sep 27th 2020)

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.

• CommentRowNumber13.
• CommentAuthorTim Campion
• CommentTimeSep 28th 2020

Thanks so much! Apparently Joyal didn’t really write anything more than is found on the nlab page.

• CommentRowNumber14.
• CommentAuthorTim Campion
• CommentTimeOct 1st 2020

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.

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeOct 2nd 2020

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.

• CommentRowNumber16.
• CommentAuthorUlrik
• CommentTimeOct 2nd 2020
• (edited Oct 2nd 2020)

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.

1. adding a definition of dagger category which does not violate the principle of equivalence

Anonymous

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeMay 4th 2022

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeMay 4th 2022

but may have missed some

2. addressing the principle of equivalence in the identity-of-objects definition of a dagger category: it depends crucially on the exact definition of identity-of-objects

Anonymous

• CommentRowNumber21.
• CommentAuthorGnampfissimo
• CommentTimeAug 23rd 2022
• (edited Aug 23rd 2022)
A few revisions ago (edit: I just saw that it was yesterday), the remark that the definition via contravariant endofunctor violates the principle of equivalence was deleted, and a paragraph was added to explain why it doesn't. However I think it's not very clearly written, e.g.:
- What is "set-valued equality for strict dagger categories"?
- What does it mean to replace equality of objects by bijection?
Furthermore I disagree with what I think I understand: even when we use type-theoretic foundations and demand equality of objects via the identity type, this still violates the principle of equivalence. An example of a similar situation is given by Grothendieck fibrations in HoTT, where we can avoid this violation by using displayed categories, where instead of demanding paths between objects, stuff is indexed by the same object (judgmentally). Applying this POV to dagger categories corresponds to the definition via a family of functions, indexed by two objects A, B.

Overall I think we should revert this change. In case you disagree with me, I'd suggest that the paragraph deserves a rewrite to be more comprehensible.
3. Edited incorrect remark on the definition of dagger category based upon identity-on-objects functors: strict categories violate the principle of equivalence, and every category in a set theory foundations is a strict category.

Anonymous

• CommentRowNumber23.
• CommentAuthorGuest
• CommentTimeAug 23rd 2022

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.

• CommentRowNumber24.
• CommentAuthormaxsnew
• CommentTimeAug 23rd 2022

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.

• CommentRowNumber25.
• CommentAuthorGuest
• CommentTimeAug 23rd 2022

@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.

4. get rid of reference to infinity-groupoid in remark: not all type theories model infinity-groupoids

Anonymous

5. expanded on the explanation for the choice of terminology

Anonymous

6. merging the paragraphs on type theory terminology into the section on terminology

Anonymous

7. editing explanation to show why the functor definition in type theory fails, and why one needs to use a displayed category instead.

Anonymous

• CommentRowNumber30.
• CommentAuthorGuest
• CommentTimeAug 23rd 2022

@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.

• CommentRowNumber31.
• CommentAuthorMike Shulman
• CommentTimeAug 29th 2022

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.

• CommentRowNumber32.
• CommentAuthorMike Shulman
• CommentTimeAug 30th 2022

A number of changes:

• Removed discussion of precategory/univalent category terminology. I don’t think this is appropriate for this entry; it should go at a renamed and generalized version of univalent dagger category.
• Corrected link to Springer EOM and concised it.
• Moved “family of functions” definition first, as I think it is the most straightforward to understand.
• Removed the lengthy explicit definition using “contravariant identity-on-objects functor”. I think this is not appropriate for this entry; it should be at identity-on-objects functor with $\dagger$-categories as an example.
• Pointed out that $F\circ \dagger$ and $\ddagger \circ F^{op}$ automatically agree on objects as soon as $\dagger$ and $\ddagger$ are the identity on objects.