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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 ?

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    $\begingroup$ 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? $\endgroup$ Jan 12 '20 at 20:42
  • $\begingroup$ 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...(?) $\endgroup$
    – nicolas
    Jan 13 '20 at 9:43
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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, and the presheaves on it are called symmetric sets, 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.

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  • $\begingroup$ I was indeed looking at a really applied case, where some presheaf seemed to make sense, to explicit some homotopy. as a category it is equivalent to FinSet because we map sets to indiscrete categories, but maybe 2-functors are more relevant here (?) $\endgroup$
    – nicolas
    Jan 12 '20 at 21:44

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