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Exercise 9.1.4.12. Let $\operatorname{\mathcal{C}}$ be the category described as follows:

  • The set of objects $\operatorname{Ob}(\operatorname{\mathcal{C}})$ is given by $\{ X_ n \} _{n \in \operatorname{\mathbf{Z}}}$.

  • Morphisms in $\operatorname{\mathcal{C}}$ are given by

    \[ \operatorname{Hom}_{ \operatorname{\mathcal{C}}}( X_ m, X_ n ) = \begin{cases} \{ 0, 1 \} & \text{ if $m \leq n$ } \\ \emptyset & \text{ otherwise, } \end{cases} \]

    with composition given by multiplication on the set $\{ 0, 1 \} $.

Then the diagram

\[ \cdots \rightarrow X_{-2} \xrightarrow {1} X_{-1} \xrightarrow {1} X_0 \xrightarrow {1} X_1 \xrightarrow {1} X_2 \xrightarrow {1} \cdots \]

determines a functor of $\infty $-categories $F: \operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}, \leq ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ which is essentially surjective, and therefore weakly right cofinal (Example 9.1.4.3). Show the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ filtered but $F$ is not right cofinal.