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1. • CommentRowNumber1.
   • CommentAuthortrent
   • CommentTimeSep 19th 2017
   • (edited Sep 19th 2017)
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   Author: trent
   Format: MarkdownItexlooking at the page for [[duality]] there is all kinds of interesting info. for [[triality]], nothing. this isn't particular to the nlab, in math at large triality is much much much less explored than duality. are there enough links between idk geoemtry/topology, physics, and number theory for people to reflect on that and achieve some sort of nontrivial understanding of triality in general? or is the development of math/physics not yet at a stage where we're prepared to have a good understanding of triality? debating asking this on mathoverflow, but haven't yet becuase I'm worried it will just get deleted as too vague/general. asking here becuase category theorists are probably the most likely group of mathematicians to reflect on something like this. edit: asked on mathoverflow too [https://mathoverflow.net/questions/281462/examples-of-triality-in-mathematics](https://mathoverflow.net/questions/281462/examples-of-triality-in-mathematics) . we'll see whether it is deleted.

   looking at the page for duality there is all kinds of interesting info. for triality, nothing. this isn’t particular to the nlab, in math at large triality is much much much less explored than duality. are there enough links between idk geoemtry/topology, physics, and number theory for people to reflect on that and achieve some sort of nontrivial understanding of triality in general? or is the development of math/physics not yet at a stage where we’re prepared to have a good understanding of triality?

   debating asking this on mathoverflow, but haven’t yet becuase I’m worried it will just get deleted as too vague/general. asking here becuase category theorists are probably the most likely group of mathematicians to reflect on something like this.

   edit: asked on mathoverflow too https://mathoverflow.net/questions/281462/examples-of-triality-in-mathematics . we’ll see whether it is deleted.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeSep 19th 2017
   • (edited Sep 19th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am wondering if there really is "triality" on par with "duality". Of course there are situations where $n$ aspects are being related, more or less precisely, for $n \geq 3$. But is there any such situation where this relation is strong enough to have the same character as the category-theoretic concept(s) of duality? You seem to suggest that > geoemtry/topology, physics, and number theory might be an example. But I am doubtful. On the one hand this is most vague. On the other hand, via arithmetic geometry we may understand "number theory" as a sector of "geometry". The only really mathematically solid phenomenon I know that rightly goes by "triality" is the one in $Spin(8)$ representation theory (e.g. [here](http://math.ucr.edu/home/baez/octonions/node7.html)), based on the fact that the Dynkin diagram of $\mathrm{Spin}(8)$ has one center vertex with three edges emerging. But, as this already indicates, this is all about a glorified version of the symmetries of triangle. This is nice, but it is not the "categorified" kind of symmetry that, in one dimension down, is called "duality". So I remain doubtful. I have yet to see a phenomenon that suggests that the category theoretic concept(s) of "duality" should generalize to "triality". What I did see examples of is "quaternality" (or whatever it should be called) in the guise of second-order dualities. For instance where an adjunction is a mathematical incarnation of duality, one may consider adjunctions in a category of adjunctions as "second order dualities". These are equivalently adjoint triples (see [here](https://ncatlab.org/nlab/show/adjoint+triple#AsAdjunctionOfAdjunctions)) and these actually do occur in nature!

   I am wondering if there really is “triality” on par with “duality”.

   Of course there are situations where $n$ aspects are being related, more or less precisely, for $n \geq 3$. But is there any such situation where this relation is strong enough to have the same character as the category-theoretic concept(s) of duality?

   You seem to suggest that

   geoemtry/topology, physics, and number theory

   might be an example. But I am doubtful. On the one hand this is most vague. On the other hand, via arithmetic geometry we may understand “number theory” as a sector of “geometry”.

   The only really mathematically solid phenomenon I know that rightly goes by “triality” is the one in $Spin(8)$ representation theory (e.g. here), based on the fact that the Dynkin diagram of $\mathrm{Spin}(8)$ has one center vertex with three edges emerging. But, as this already indicates, this is all about a glorified version of the symmetries of triangle. This is nice, but it is not the “categorified” kind of symmetry that, in one dimension down, is called “duality”.

   So I remain doubtful. I have yet to see a phenomenon that suggests that the category theoretic concept(s) of “duality” should generalize to “triality”.

   What I did see examples of is “quaternality” (or whatever it should be called) in the guise of second-order dualities.

   For instance where an adjunction is a mathematical incarnation of duality, one may consider adjunctions in a category of adjunctions as “second order dualities”. These are equivalently adjoint triples (see here) and these actually do occur in nature!

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 19th 2017
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   Author: David_Corfield
   Format: MarkdownItexI guess if you see duality as importantly relating its two guises, arrow reversal and dualizing object, it's hard to see how there will be an equivalent for triality. > Not every statement will be taken into its formal dual by the process of dualizing with respect to $V$, and indeed a large part of the study of mathematics >>space vs. quantity > and of logic >>theory vs. example >may be considered as the detailed study of the extent to which formal duality and concrete duality into a favorite $V$ correspond or fail to correspond. (Lawvere and Rosebrugh) What would replace arrow reversal? And where a [[dualizing object]], $D$, reverses direction, $A \to B \mapsto D^B \to D^A$, a 'trializing' object would have the wrong parity.

   I guess if you see duality as importantly relating its two guises, arrow reversal and dualizing object, it’s hard to see how there will be an equivalent for triality.

   Not every statement will be taken into its formal dual by the process of dualizing with respect to $V$, and indeed a large part of the study of mathematics

   space vs. quantity

   and of logic

   theory vs. example

   may be considered as the detailed study of the extent to which formal duality and concrete duality into a favorite $V$ correspond or fail to correspond. (Lawvere and Rosebrugh)

   What would replace arrow reversal? And where a dualizing object, $D$, reverses direction, $A \to B \mapsto D^B \to D^A$, a ’trializing’ object would have the wrong parity.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeSep 19th 2017
   • (edited Sep 19th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes. Of course people might argue that this is just reflecting our inability to look beyond the established mathematics of duality. But maybe the phenomenon of [[adjoint triples]] has not found the kind of attention yet, among matheo-philosophically inclined people, that they deserve. Maybe much of the desire for triality can find a satisfactory answer here.

   Yes. Of course people might argue that this is just reflecting our inability to look beyond the established mathematics of duality.

   But maybe the phenomenon of adjoint triples has not found the kind of attention yet, among matheo-philosophically inclined people, that they deserve. Maybe much of the desire for triality can find a satisfactory answer here.

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 19th 2017
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   Author: David_Corfield
   Format: MarkdownItexWhat, some kind of triangular [[hyperstructures]]?

   What, some kind of triangular hyperstructures?

6. • CommentRowNumber6.
   • CommentAuthorNoam_Zeilberger
   • CommentTimeSep 19th 2017
   • (edited Sep 19th 2017)
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   Author: Noam_Zeilberger
   Format: MarkdownItexBill Tutte wrote a paper called "Duality and Trinity", which was about how the duality relationship for embedded graphs can be generalized to a triality relationship for embedded bipartite graphs/hypergraphs. (There is a nice exposition in [this article](https://arxiv.org/pdf/1702.03630.pdf) by Daniel Mathews.) Trivalent graphs also seem to play an important role in graph theory. Besides the adjoint triples Urs mentioned, some other places you might see "trialities" working in category theory are: 1. [[Monoidal biclosed categories]] $\mathcal{V}$: an equivalence $\mathcal{V}(A \otimes B,C) \cong \mathcal{V}(A, C / B) \cong \mathcal{V}(B, A \setminus C)$ for any triple of objects $A, B, C \in \mathcal{V}$; 1. [[Bifibrations]] $\mathcal{E} \to \mathcal{B}$: an equivalence $\mathcal{E}_f(R,S) \cong \mathcal{E}_A(R,pull_f(S))\cong \mathcal{E}_B(push_f(R),S) $ for any morphism $f : A \to B \in \mathcal{B}$ and pair of objects $R \in \mathcal{E}_A$ and $S \in \mathcal{E}_B$ (writing $\mathcal{E}_f(R,S)$ for the set of morphisms from $R$ to $S$ in $\mathcal{E}$ lying over $f$).

   Bill Tutte wrote a paper called “Duality and Trinity”, which was about how the duality relationship for embedded graphs can be generalized to a triality relationship for embedded bipartite graphs/hypergraphs. (There is a nice exposition in this article by Daniel Mathews.) Trivalent graphs also seem to play an important role in graph theory.

   Besides the adjoint triples Urs mentioned, some other places you might see “trialities” working in category theory are:

   1. Monoidal biclosed categories $\mathcal{V}$: an equivalence $\mathcal{V}(A \otimes B,C) \cong \mathcal{V}(A, C / B) \cong \mathcal{V}(B, A \setminus C)$ for any triple of objects $A, B, C \in \mathcal{V}$;
   2. Bifibrations $\mathcal{E} \to \mathcal{B}$: an equivalence $\mathcal{E}_f(R,S) \cong \mathcal{E}_A(R,pull_f(S))\cong \mathcal{E}_B(push_f(R),S)$ for any morphism $f : A \to B \in \mathcal{B}$ and pair of objects $R \in \mathcal{E}_A$ and $S \in \mathcal{E}_B$ (writing $\mathcal{E}_f(R,S)$ for the set of morphisms from $R$ to $S$ in $\mathcal{E}$ lying over $f$).
7. • CommentRowNumber7.
   • CommentAuthormaxsnew
   • CommentTimeSep 19th 2017
   • (edited Sep 19th 2017)
   • PermaLink
   Author: maxsnew
   Format: MarkdownItexWe can also generalize Noam's examples > [[Monoidal biclosed categories]] is an instance of a [[two-variable adjunction]], and this generalizes to [multivariable adjunctions](https://arxiv.org/abs/1208.4520) so you get 4-alities or whatever too. Whereas > [[Bifibrations]] is an instance of the relationship between a profunctor and an adjoint pair that represent it. Applying this to the monoidal biclosed categories, you would actually get a "4-ality": primitive multicategory hom and then 3 representations $\mathcal{V}(A,B;C) \cong \mathcal{V}(A \otimes B;C) \cong \mathcal{V}(A; C / B) \cong \mathcal{V}(B; A \setminus C)$.

   We can also generalize Noam’s examples

   Monoidal biclosed categories

   is an instance of a two-variable adjunction, and this generalizes to multivariable adjunctions so you get 4-alities or whatever too.

   Whereas

   Bifibrations

   is an instance of the relationship between a profunctor and an adjoint pair that represent it.

   Applying this to the monoidal biclosed categories, you would actually get a “4-ality”: primitive multicategory hom and then 3 representations $\mathcal{V}(A,B;C) \cong \mathcal{V}(A \otimes B;C) \cong \mathcal{V}(A; C / B) \cong \mathcal{V}(B; A \setminus C)$.

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeSep 19th 2017
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   Author: Mike Shulman
   Format: MarkdownItexSomething that one might argue generalizes "arrow reversal" to $n\gt 2$ is the structure of a cyclic multicategory, in which a multi-arrow $(A_1,\dots,A_n) \to B$ can have its input and output cyclically permuted to become $(A_2,\dots,A_n,B^\bullet) \to A_1^\bullet$ (where perhaps $A^\bullet = A$ or perhaps not, depending on the author). In particular, multivariable adjunctions form a cyclic multicategory, with $A^\bullet = A^{op}$. Also, did anyone mention yet the fact that $n$-categories have $2^n-1$ "opposites"?

   Something that one might argue generalizes “arrow reversal” to $n\gt 2$ is the structure of a cyclic multicategory, in which a multi-arrow $(A_1,\dots,A_n) \to B$ can have its input and output cyclically permuted to become $(A_2,\dots,A_n,B^\bullet) \to A_1^\bullet$ (where perhaps $A^\bullet = A$ or perhaps not, depending on the author). In particular, multivariable adjunctions form a cyclic multicategory, with $A^\bullet = A^{op}$.

   Also, did anyone mention yet the fact that $n$-categories have $2^n-1$ “opposites”?

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeSep 19th 2017
   • PermaLink
   Author: Urs
   Format: MarkdownItexGood points. So a 2-variable (or higher) adjunction is an instance of iterated duality, not unlike the case of adjoint triples, where $A$ is dual to $B$ is dual to $C$. Do we want to say that $n$-alities are sequences of dualities?

   Good points. So a 2-variable (or higher) adjunction is an instance of iterated duality, not unlike the case of adjoint triples, where $A$ is dual to $B$ is dual to $C$. Do we want to say that $n$-alities are sequences of dualities?

10. • CommentRowNumber10.
    • CommentAuthortrent
    • CommentTimeSep 19th 2017
    • PermaLink
    Author: trent
    Format: MarkdownItex> Do we want to say that n-alities are sequences of dualities? I didn't bother asking at first because I figured "i don't even know what a triality is yet, I have no business worrying about anything higher", but... now that n-alities have come up: I hope there is a way to formulate the concept of -ality such that you have indecomposable p-alities for all primes p. supposing that isn't possible though, a sequence of dualities seems the most natural option. (and even if p-alities are possible, I'm sure there are still practical situations in math and physics where the more trivial notion of iterated duality arises).

    Do we want to say that n-alities are sequences of dualities?

    I didn’t bother asking at first because I figured “i don’t even know what a triality is yet, I have no business worrying about anything higher”, but… now that n-alities have come up: I hope there is a way to formulate the concept of -ality such that you have indecomposable p-alities for all primes p. supposing that isn’t possible though, a sequence of dualities seems the most natural option. (and even if p-alities are possible, I’m sure there are still practical situations in math and physics where the more trivial notion of iterated duality arises).

11. • CommentRowNumber11.
    • CommentAuthortrent
    • CommentTimeSep 19th 2017
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    Author: trent
    Format: MarkdownItexOn second thought, an issue I have with triality treated simply as a sequence of dualities is that if we know a is dual to b (without reference to c) and b is dual to c (without reference to a), but cannot prove that a is dual to c expect with reference to dualities between a/b and b/c, then I would consider that a degenerate triality. One could make this non-degenerate in two directions: 1. by requiring a reference to all three terms to establish any particular duality 2. by having proofs of duality between any two terms that only reference those two terms. The first option is the more interesting of the two since there is something irreducibly triadic there.

    On second thought, an issue I have with triality treated simply as a sequence of dualities is that if we know a is dual to b (without reference to c) and b is dual to c (without reference to a), but cannot prove that a is dual to c expect with reference to dualities between a/b and b/c, then I would consider that a degenerate triality. One could make this non-degenerate in two directions:

    1. by requiring a reference to all three terms to establish any particular duality

    2. by having proofs of duality between any two terms that only reference those two terms.

    The first option is the more interesting of the two since there is something irreducibly triadic there.