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Stream: practice: terminology & notation

Topic: 3-dimensional opposite

Matteo Capucci (he/him) (Jun 27 2024 at 09:55):

Is there an established name for the operation that reverses the 3-cells of a tricategory?

John Baez (Jun 27 2024 at 10:04):

Not that I've heard. Sometime around around $n = 3$ people start writing $\mathrm{op}_1, \mathrm{op}_2, \dots, \mathrm{op}_n$. But I agree that if you're doing a lot of tricategorical stuff it would be nice to have something after "op" and "co". What would be good?

Matteo Capucci (he/him) (Jun 27 2024 at 10:07):

Indeed. I was considering 'du' as in 'dual', for lack of a better alternative...

Amar Hadzihasanovic (Jun 27 2024 at 10:08):

For some good reasons, it is becoming more established among $n$-categorists that $-^\mathrm{op}$ reverses cells in all odd dimensions while $-^\mathrm{co}$ reverses cells in all even dimensions, so I would recommend following this convention.

Matteo Capucci (he/him) (Jun 27 2024 at 10:08):

wait, all even/odd dimensions at once?

Amar Hadzihasanovic (Jun 27 2024 at 10:08):

Yes, so when dealing with tricategories you shouldn't use $-^\mathrm{op}$ unless both 1-cells and 3-cells are reversed.

Matteo Capucci (he/him) (Jun 27 2024 at 10:09):

What's the motivation? Is it evil to reverse only one level?

Matteo Capucci (he/him) (Jun 27 2024 at 10:10):

Funnily enough in my current use case I do indeed need to reverse both 1- and 3-cells... what sorcery is this

Amar Hadzihasanovic (Jun 27 2024 at 10:10):

The main reason is that they are the only dualities that are compatible with Gray products (up to a twist).

Matteo Capucci (he/him) (Jun 27 2024 at 10:11):

I see

Matteo Capucci (he/him) (Jun 27 2024 at 10:11):

Cool!

Amar Hadzihasanovic (Jun 27 2024 at 10:12):

Also $-^\mathrm{op}$ is compatible with n-categorical joins (of which cones are an example) up to a twist, so e.g. Street's oriented simplices are stable under $-^\mathrm{op}$ and no other duality!

Vincent R.B. Blazy (Jun 27 2024 at 10:12):

@Matteo Capucci (he/him) Did he (Gray) already show up in your current use case? :grinning_face_with_smiling_eyes:

Vincent R.B. Blazy (Jun 27 2024 at 10:12):

Or 3-joins then??

Vincent R.B. Blazy (Jun 27 2024 at 10:13):

(I’m flexible in my wonderings)

Matteo Capucci (he/him) (Jun 27 2024 at 10:15):

No none of this... I'm just dualizing a construction on a 2-category that yields a trifunctor. Now if I take the $\rm co$ of the starting 2-category, I get the $\rm op_3$ of the tricategories involved by the trifunctor. But then it turns out I really want to also op the 1-cells, so I end up with the odd $\rm op$!

Matteo Capucci (he/him) (Jun 27 2024 at 10:17):

It might be some deep reasons are at work here, and that eventually one can show the two symmetries are indeed related

Vincent R.B. Blazy (Jun 27 2024 at 10:42):

Okay, yeah indeed! Maybe @Amar Hadzihasanovic will already have a clue hopefully?

Amar Hadzihasanovic (Jun 27 2024 at 13:23):

All I can guess is that, if your object can be represented as some sub-3-category of some 3-category of "functors, lax natural transformations, lax modifications...", so an internal hom right adjoint to a lax Gray product, then it should be expected that any self-duality would only feature those direction-reversals that work well with Gray products

Amar Hadzihasanovic (Jun 27 2024 at 13:25):

Same thing if your object can be represented as some sub-3-category of a "lax slice" or "colax slice", since slices are locally right adjoint to joins

Amar Hadzihasanovic (Jun 27 2024 at 13:27):

A lot of functorial constructions in category theory involve either slicing or functor categories, and it should be expected that the same is true in higher dim

Matteo Capucci (he/him) (Jun 27 2024 at 13:38):

uhm indeed, now that I think about it, it's a duality regarding pseudocategories in a 2-category, which are a particular case of lax functors into Span(K). but K^co only gives me $PsCat(K)^{op_3}$, the 1-opposite is something I do for aesthetic reasons... basically.

Matteo Capucci (he/him) (Jun 27 2024 at 13:51):

Amar, what about presheaves? Would one still look at $K^{op} \to \bf nCat$? Perhaps a better question is, what about fibrations? Perhaps in that case one looks at $K^{op,co}$ (a 2-fibration has cartesian lifts for both 1- and 2-cells)

Amar Hadzihasanovic (Jun 27 2024 at 15:35):

I don't know, sorry, I haven't really ever given much thought to indexed higher cats

David Kern (Jun 28 2024 at 23:19):

Yes, it should be the total dual $K^{\operatorname{op},\operatorname{co}}$: for example, Loubaton defines in Section 3.2 fibrations (aka right cartesian fibrations, aka cartesian fibrations) of $(\infty,\omega)$-categories as right orthogonal to initial functors, and opfibrations (aka left cartesian, aka cocartesian, fibrations) as right orthogonal to final functors, and notes that total duality exchanges the classes of initial and final functors (the link between his definition of initial/final functors and more usual ones is Proposition 3.2.5.21 and Corollary 4.2.3.22).
Another really nice way of seeing how this dualisation fits in is to use the characterisation of op/fibrations from Theorem 3.2.2.24 (5) and (5)', saying that a functor is a fibration (reps. opfibration) if and only if its has cartesian (resp. cocartesian) liftings of 1-cells and for any pair of objects the induced functor on hom $(\infty,\omega)$-categories is an opfibration (resp. a fibration) — note that in the finite case, for $(\infty,\ell)$-categories with $\ell<\omega$, this coinductive characterisation reduces to the inductive definition given by Nuiten (Definition 5.10, and cf. Remarks 5.14 and 5.15 for the dualisations). This shows that the two notions of fibrations and opfibrations are closely entangled together in a way that exactly reflects the even/odd pattern that Amar was talking about, so that the way to exchange them is by full duality.

Matteo Capucci (he/him) (Jun 30 2024 at 07:36):

Thanks David. I'm not following the second part of your message: a 2-fibration according eg to Hermida is locally still a fibration, not a opfibration? What am I missing?

David Kern (Jun 30 2024 at 17:45):

Trying to compare Joost's explanation (N.B.: he does mention in the introduction that he has a different variance than Hermida) with the one in paragraph 4 of the nLab page [[n-fibration]], I am led to conclude that the difference is just that they take slightly different Grothendieck constructions. Indeed, if you try to reproduce the explanation on the nLab assuming that your $p$ is locally an opfibration, when they take a cartesian $2$-cell $\psi=\alpha^\ast\phi\Rightarrow\phi$ and factor $\psi$ through the cartesian $1$-cell $\chi\colon f^\ast a\to a$, you can instead take a cocartesian $2$-cell $\phi\Rightarrow\alpha_!\phi$ and factor $\alpha_!\phi\colon g^\ast a\to a$ in the same way through $\chi$ to also get an arrow $g^\ast a\to f^\ast a$ with the desired variance.
So I suppose that the reason Nuiten and Loubaton both use the definition of fibration with mixed variance is that it must somehow be easier to handle — at least $\infty$-categorically, where you can't do the Grothendieck construction by hand — than the homogeneous one, but either way full duality will always be what exchanges fibrations and opfibrations.

Matteo Capucci (he/him) (Jul 02 2024 at 08:04):

I ended up using $(-)^{\rm re}$ in the end. The duality turned out to not involve 1-cells :(