# Symetrical simplex category

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 ?

• ncatlab.org/nlab/show/indiscrete+category. This construction is so general and ignores all the interesting aspects of simplices that it is useless. Is it really what you want to ask? Jan 12, 2020 at 20:42
• I see what you mean. up to iso of course. Physicists like to "degenerate" things to have a better understanding by gradually adding finer features : to go from this to Simp is a smaller step than to go from nothing to Simp, with all its goodness (nerve and all ...), though it might be just a waste in that case...(?) Jan 13, 2020 at 9:43