Warning 9.2.9.5 (068Z)

Warning 9.2.9.5. In the situation of Theorem 9.2.9.4, the assumption $\lambda \trianglelefteq \mu $ cannot be omitted. For example, let $S$ be set of cardinality $\geq \mu $, let $P_{< \kappa }(S)$ denote the nerve of the partially ordered set of all $\kappa $-small subsets of $S$, and define $P_{< \lambda }(S)$ and $P_{< \mu }(S)$ similarly. Assume that $\kappa \trianglelefteq \lambda $ and $\kappa \trianglelefteq \mu $ (this condition is always satisfied, for example, if $\kappa = \aleph _0$: see Example 9.1.7.10). Then Example 9.2.3.11 supplies equivalences

\[ \operatorname{Ind}_{\kappa }^{\lambda }( P_{< \kappa }(S) ) \xrightarrow {\sim } P_{< \lambda }(S) \quad \quad \operatorname{Ind}_{\kappa }^{\mu }( P_{< \kappa }(S) ) \xrightarrow {\sim } P_{< \mu }(S). \]

In this case, the conclusion of Theorem 9.2.9.4 is satisfied if and only if the comparison map $\operatorname{Ind}_{\lambda }^{\mu }( P_{< \lambda }(S) ) \hookrightarrow P_{< \mu }(S)$ is an equivalence, which is equivalent to the requirement $\lambda \trianglelefteq \mu $ (Example 9.2.3.11).