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Example 1.4.1.7. Let $X$ be a topological space and let $\operatorname{\mathcal{C}}$ be an ordinary category. To specify a functor of $\infty $-categories $F: \operatorname{Sing}_{\bullet }(X) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, one must give a rule which assigns to each continuous map $\sigma : | \Delta ^ n | \rightarrow X$ (viewed as an $n$-simplex of $\operatorname{Sing}_{\bullet }(X)$) a diagram $F(\sigma ) = ( C_0 \xrightarrow {f_1} C_1 \xrightarrow {f_2} C_2 \rightarrow \cdots \xrightarrow {f_ n} C_ n)$. In particular:

$(a)$

To each point $x \in X$, the functor $F$ assigns an object $F(x) \in \operatorname{\mathcal{C}}$.

$(b)$

To each continuous path $f: [0,1] \rightarrow X$ starting at the point $x = f(0)$ and ending at the point $y = f(1)$, the functor $F$ assigns a morphism $F(f): F(x) \rightarrow F(y)$ in the category $\operatorname{\mathcal{C}}$. The morphism $F(f)$ is automatically an isomorphism (by virtue of Proposition 1.3.6.10 and Remark 1.4.1.6).

$(c)$

For each continuous map $\sigma : | \Delta ^2 | \rightarrow X$ with boundary behavior as depicted in the diagram

\[ \xymatrix { & y \ar [dr]^{g} & \\ x \ar [ur]^{f} \ar [rr]^{h} & & z, } \]

we have an identity $F(h) = F(g) \circ F(f)$ in $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(x), F(z) )$.

The data of a collection of objects $\{ F(x) \} _{x \in X}$ and isomorphisms $\{ F(f) \} _{ f: [0,1] \rightarrow X}$ satisfying $(c)$ is called a $\operatorname{\mathcal{C}}$-valued local system on $X$. The preceding discussion determines a bijection

\[ \xymatrix { \{ \text{Functors of $\infty $-categories from $\operatorname{Sing}_{\bullet }(X)$ to $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$} \} \ar [d]^{\sim } \\ \{ \text{ $\operatorname{\mathcal{C}}$-valued local systems on $X$} \} . } \]

By virtue of Example 1.4.1.2, we can also identify local systems with functors from the fundamental groupoid $\pi _{\leq 1}(X)$ into $\operatorname{\mathcal{C}}$.