Groupoid objects

Let \(C\) be a category. A simplicial object \(\Gg\colon\simp\catop \to C\) is called a groupoid object if for every \([n] \in \simp\) and every (not necessarily order-preserving) partition \begin{align*} [n] \simeq \{i_0,\dots ,i_k\} \sqcup_{\{i_k\}} \{i_k,\dots,i_n\}, \end{align*} the induced diagram

is a pullback square in \(C\). We let \(\Grpd(C) \subseteq \Fun(\simp\catop,C)\) denote the full subcategory spanned by the groupoid objects in \(C\).