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.

Author: mathtuition88

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