Definition 3.1.5.1. Let $X_{}$ and $Y_{}$ be simplicial sets, and suppose we are given a pair of maps $f,g: X_{} \rightarrow Y_{}$, which we identify with vertices of the simplicial set $\operatorname{Fun}( X_{}, Y_{} )$. We will say that $f$ and $g$ are homotopic if they belong to the same connected component of the simplicial set $\operatorname{Fun}(X_{}, Y_{} )$ (Definition 1.2.1.8).
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