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Evan Patterson (Jan 15 2021 at 06:16):I am trying to learn the rudiments of discrete calculus, and I would like to understand the combinatorial analogue of the oriented simplicial complexes that show up there. I think this concept should be intermediate between simplicial sets, where the faces are fully ordered, and symmetric simplicial sets, where an action of the symmetric group at every dimension effectively renders the faces unordered. Since the alternating group is the group of orientation-preserving symmetries, I imagine that the combinatorial analogue of an oriented simplicial complex should be an "alternating simplicial set," a presheaf over the simplicial category extended with an action of the alternating group at every dimension.
As someone who doesn't know much about simplicial stuff, I am wondering: does this make sense? Is it a standard concept? Any relevant references?
Tim Hosgood (Jan 15 2021 at 17:38):according to that nlab page, the concept in between simplicial sets and symmetric ones is that of “cyclic sets”
Tim Hosgood (Jan 15 2021 at 17:39):(that’s not at all an answer to your question, but just thought it worth pointing out) (i’d also like to see an actual answer to your question!)
John Baez (Jan 15 2021 at 19:16):Evan Patterson said:
I am trying to learn the rudiments of discrete calculus, and I would like to understand the combinatorial analogue of the oriented simplicial complexes that show up there. I think this concept should be intermediate between simplicial sets, where the faces are fully ordered, and symmetric simplicial sets, where an action of the symmetric group at every dimension effectively renders the faces unordered. Since the alternating group is the group of orientation-preserving symmetries, I imagine that the combinatorial analogue of an oriented simplicial complex should be an "alternating simplicial set," a presheaf over the simplicial category extended with an action of the alternating group at every dimension.
As someone who doesn't know much about simplicial stuff, I am wondering: does this make sense? Is it a standard concept? Any relevant references?
I've never heard of this concept. To make it precise one needs to figure out how to combine even permutations with face and degeneracy maps into a category that doesn't include all maps between finite sets.
Simplicial sets are presheaves on the category of finite totally ordered nonempty sets and order-preserving maps. (We usually use a skeleton of this category but from a certain viewpoint that's not such a big deal.)
Symmetric simplical sets are presheaves on the category of nonempty finite sets and all maps.
You seem to want a category of oriented nonempty finite sets and orientation-preserving (or "even") maps. I don't know if such a category exists.
An orientation on a finite set is, roughly, a choice of which total orderings of that set count as "right-handed" and which count as "left-handed", subject to the consistency condition that an even permutation applied to ordering preserves its handedness, while an odd permutation reverses it. You can read about a similar concept for vector spaces here:
The concept of orientation for finite sets is for some reason less widely known.
There's a perfectly fine groupoid of oriented nonempty finite sets and orientation-preserving bijections. This has as a skeleton the groupoid where the objects are n = 1, 2, 3, ... and the morphisms, all automorphisms, are elements of the alternating groups .
The question becomes: which maps between oriented sets of different cardinalities count as orientation-preserving??? I suspect there's no good answer to this, and thus no good answer to your original question. But I haven't proved it.
Evan Patterson (Jan 15 2021 at 20:18):Thanks Tim and John for the helpful perspectives. I agree that there is something puzzling, if not downright problematic, about what the maps between oriented sets of different sizes should be. I am mostly interested in the face maps, so if it helps we can forget about the degeneracies and think about semi-simplicial sets.
Matteo Capucci (he/him) (Jan 16 2021 at 11:52):There are some nice answers here: https://math.stackexchange.com/questions/5191/what-structure-does-the-alternating-group-preserve
John Baez (Jan 16 2021 at 17:11):I guess one big question, when attempting to define a category rather than a mere groupoid of oriented finite sets, is whether you want to include the unique map in this category or not.