The simplex category has for objects totally ordered sets $[n]$ , and for morphisms order-preserving functions between those sets.


We can see the totally ordered set $[n]$ of size $n$ of the simplex category as a very simple form of category (skeletal), for which between 2 elements, there is **at most** one arrow, which witnesses the fact that X <= Y.
By anti-symetry, if X < Y, witnessed by an element in $Hom[n](X,Y)$ then the set $Hom[n](Y,X)$ is empty since X <> Y. By totality, either one homset has one element.


(Interestingly, viewing ordered sets as a basic form of (skeletal) category, then the simplex category is a basic form of the 2-category $Cat$.)


Now, what if we have, instead of those skeletal categories $[n]$, that between two objects X and Y, there is always a pair of arrow, in opposite direction, *inverse* of each other, so that, calling XY what was previously witnessing the fact that X < Y, XY = -YX. This is still a category, even simpler to describe : with **exactly** one morphism in each homset. period.


And my question is : does this "symetric simplex" category has a name ?



Symetrical simplex category
You haven't specified what the *morphisms* in your "symmetric simplex category" are supposed to be, and there are two natural choices:


* Functors, or
* Natural isomorphism classes of functors.


In the first case, you get a category equivalent to [FinSet](https://ncatlab.org/nlab/show/FinSet), and the presheaves on it are called [symmetric sets](https://ncatlab.org/nlab/show/symmetric+set), and AFAIK these also model homotopy theory.


In the second case, all your categories are equivalent, so that your "symmetric simplex category" itself has exactly one morphism in each direction between any two objects, i.e. it is indiscrete and therefore equivalent to the terminal category, which (as Martin says) makes it uninteresting.


