simplicial sets with symmetries ?

[Bas Spitters]

Cubical sets come in many variants. What is known about variations on
simplicial sets?
Specifically, have simplicial sets with symmetries have been studied?

I expect them to classify linear orders with involution.
I don't think it would be very difficult to define them explicitly,
but I would be more interested if this has already been studied.

There seem to be two related shape categories:
* the crossed simplicial group Z_2, example 1 in:
http://www.ams.org/journals/tran/1991-326-01/S0002-9947-1991-0998125-4/home.html
http://ncatlab.org/nlab/show/skew-simplicial+set

* Dagger simplicial sets by Joyal:
http://ncatlab.org/nlab/show/dagger-category#simplicial_set
http://permalink.gmane.org/gmane.science.mathematics.categories/5477

However, this seems to be a linear order which "forgets" the
orientation, as in the
examples of Connes cyclic sets (which classify abstract circles) and
symmetric sets. This seems different from a linear order with an
explicit involution.

Thanks,

Bas

[Steve Awodey]

> On Aug 23, 2015, at 7:56 AM, Bas Spitters <b.a.w.s...@gmail.com> wrote:
>
> Cubical sets come in many variants. What is known about variations on
> simplicial sets?
> Specifically, have simplicial sets with symmetries have been studied?

these are standard in alg. top. and are called symetric simplicial sets.
the nLab calls them "symmetric sets”, which is a bit odd, but defines them corectly as presheaves on the finite, non-empty sets.
I guess this classifies “non-degenerate” boolean algebras, i.e. with 0 /= 1.

Steve

>
> I expect them to classify linear orders with involution.
> I don't think it would be very difficult to define them explicitly,
> but I would be more interested if this has already been studied.
>
> There seem to be two related shape categories:
> * the crossed simplicial group Z_2, example 1 in:
> http://www.ams.org/journals/tran/1991-326-01/S0002-9947-1991-0998125-4/home.html
> http://ncatlab.org/nlab/show/skew-simplicial+set
>
> * Dagger simplicial sets by Joyal:
> http://ncatlab.org/nlab/show/dagger-category#simplicial_set
> http://permalink.gmane.org/gmane.science.mathematics.categories/5477
>
> However, this seems to be a linear order which "forgets" the
> orientation, as in the
> examples of Connes cyclic sets (which classify abstract circles) and
> symmetric sets. This seems different from a linear order with an
> explicit involution.
>
> Thanks,
>
> Bas
>

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