ScalingStacks

A.2.4 Simplicial objects[00IE]

We write \(\Delta\) for the simplicial indexing category (the full subcategory of \(\mathrm{Cat}_\infty\) on the finite nonempty totally ordered sets), and for any \(n \geq 0\) we write \([n] \coloneqq \{ 0 < 1 < \cdots < n \} \in \Delta\) for the indicated standard object. A simplicial object in an \(\infty\)-category \(\mathcal C\) is a functor \(\Delta^\mathrm{op}\xrightarrow{X} \mathcal C\); we use the term geometric realization to refer to its colimit, and denote this by \(|X| \coloneqq \mathrm{colim}_{\Delta^\mathrm{op}}(X) \in \mathcal C\).⁴⁵