Definition 1.1.3.1. Let $S$ be a simplicial set and let $k$ be an integer. We will say that $S$ has dimension $\leq k$ if every $n$-simplex of $S$ is degenerate for $n > k$. If $k \geq 0$, we say that $S$ has dimension $k$ if it has dimension $\leq k$ but does not have dimension $\leq k-1$. We say that $S$ is finite-dimensional if it has dimension $\leq k$ for some $k \gg 0$.
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