Minimal Simplicial Sets

Let X be a space. We can have a fibrant simplicial set, namely the singular simplicial set S_*(X), where S_n(x) is the set of all continuous maps from the n-simplex to X. However S_*(X) seems too large as there are uncountably many elements in each S_n(X). On the other hand, we need the fibrant assumption to have simplicial homotopy groups. This means that the simplicial model for a given space cannot be too small. We wish to have a smallest fibrant simplicial set which will be the idea behind minimal simplicial sets.

Let X be a fibrant simplicial set. For x,y\in X_n we say that x\simeq y if the representing maps f_x and f_y are homotopic relative to \partial\Delta[n].

A fibrant simplicial set is said to be minimal if it has the property that x\simeq y implies x=y.

Let X be a fibrant simplicial set. X is minimal iff for any 0\leq k\leq n+1, v,w\in X_{n+1} such that d_iv=d_iw for all i\neq k implies d_kv=d_kw.

In other words, it means that a fibrant simplicial set is minimal iff for any two elements with all faces but one the same, then the missed face must be the same.

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