Dimension of anima slice

Let \(X \in \An\). If \(X\) is represented by a \(d\)-dimensional CW-complex, or by a \(d\)-dimensional simplicial set, then \(\dim(\An_{/X}) \leq d\).

More precisely, \(\dim(\An_{/X}) \leq d\) if and only if \(X\) is a retract, in the homotopy category of animae, of an anima represented by a \(d\)-dimensional CW-complex. If \(d \neq 2\), this is further equivalent to \(X\) itself admitting a \(d\)-dimensional CW-model. For \(d=2\), the retract condition implies that \(X\) admits a \(3\)-dimensional CW-model, but it need not admit a \(2\)-dimensional one. See [Lurie 2009, Example 7.2.1.4].