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.

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, with exactly one morphism in each homset.

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

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 ?

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.

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, with additional properties.

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

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.

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, with exactly one morphism in each homset.

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