## Homotopy Theory on Simplicial Sets

Let $f,g:X\to Y$ be simplicial maps. We say that $f$ is homotopic to $g$ (denoted by $f\simeq g$) if there exists a simplicial map $F:X\times I\to Y$ such that $F(x,0)=f(x)$ and $F(x,1)=g(x)$ for all $x\in X$. If $A$ is a simplicial subset of $X$ and $f,g:X\to Y$ are simplicial maps such that $f|_A=g|_A$, we say that $f\simeq g\ \text{rel}\ A$ if there is a homotopy $F:X\times I\to Y$ such that $F(x,0)=f(x)$, $F(x,1)=g(x)$ and $F(a,t)=f(a)$ for all $x\in X$, $a\in A$, $t\in I$.

Let $X$ be a simplicial set. The elements $x_0,\dots,x_{i-1},x_{i+1},\dots,x_n\in X_{n-1}$ are said to be matching faces with respect to $i$ if $d_jx_k=d_kx_{j+1}$ for $j\geq k$ and $k,j+1\neq i$.

A simplicial set $X$ is said to be fibrant (or Kan complex) if it satisfies the following homotopy extension condition for each $i$:

Let $x_0,\dots,x_{i-1},x_{i+1},\dots,x_n\in X_{n-1}$ be any elements that are matching faces with respect to $i$. Then there exists an element $w\in X_n$ such that $d_jw=x_j$ for $j\neq i$.

We define $\Lambda^i[n]$ as the simplicial subset of $\Delta[n]$ generated by all $d_j\sigma_n$ for $j\neq i$, where $\sigma_n=(0,1,\dots\,n)\in\Delta[n]_n$ is the nondegenerate element.

Proposition: Let $X$ be a simplicial set. Then $X$ is fibrant if and only if every simplicial map $f:\Lambda^i[n]\to X$ has an extension for each $i$.