关于单纯集到广群的态射确定性的推广问题

Hey, great question! This gets at a key property of groupoids (as 2-truncated ∞-groupoids), and your hunch about using orientations to pick edges is totally on the right track. Let's break this down:

First, let's recap the context you already laid out: for the standard 2-simplex Δ², any horn Λᵢ² (a 1-dimensional subcomplex) is enough to determine all morphisms Δ² → G. This works because groupoids have trivial higher homotopy groups—meaning any "commutative triangle" (the 2-simplex's image in G) is uniquely determined by two of its edges (and the vertices), since the third edge is just the composite (or inverse composite, depending on orientation) of the other two.

The Short Answer #

Yes! For any simplicial complex X (coming from an oriented simplicial complex, as you assumed), there does exist a 1-dimensional subcomplex Y ⊂ X such that Hom(X,G) = Hom(Y,G).

Why This Works (and How to Construct Y) #

The core idea ties back to groupoids being 2-truncated: all higher-dimensional "fillings" (for simplices of dimension ≥2) are uniquely determined by lower-dimensional data. Here's a concrete way to build Y using the orientation-induced vertex ordering you thought of:

• Start with all vertices: Include every vertex of X in Y—morphisms have to map vertices to objects of G, so we can't skip these.
• Select a generating set of edges:
  1. Use the orientation of X to fix a total order on its vertices: v₀ < v₁ < v₂ < ....
  2. For each simplex σ of X (of any dimension ≥2) with vertices vᵢ₀ < vᵢ₁ < ... < vᵢₖ, we only need to keep edges connecting consecutive vertices in this order (e.g., vᵢ₀ → vᵢ₁, vᵢ₁ → vᵢ₂, etc.). Any "jump" edge (like vᵢ₀ → vᵢ₂) can be uniquely determined by composing the consecutive edges via the commutative condition imposed by the 2-simplex spanning those three vertices.
  3. More formally, we're picking a set of edges that generates the fundamental groupoid of X. Since morphisms from X to G correspond exactly to functors from X's fundamental groupoid to G, specifying the functor on the generating objects (vertices) and generating morphisms (our selected edges) is enough to determine the entire functor.

A Quick Check with Higher Dimensions #

What about simplices of dimension ≥3? Since G is a groupoid (2-truncated), any higher-dimensional simplex's image in G is automatically determined by its 2-skeleton. And since we've already ensured the 2-skeleton's morphisms are determined by Y, this means all higher-dimensional data falls into place uniquely.

In short, your intuition about using orientation to prune edges is valid—by keeping only a generating set of edges (relative to the groupoid's commutative rules), you get a 1-dimensional subcomplex that captures all the necessary information to determine any morphism from X to G.

备注：内容来源于stack exchange，提问作者JLA