Definition 4.1.8 (Pointed objects). If \(C\) is an \(\infty \)-category with a terminal object \(*\), we define its \(\infty \)-category of pointed objects \(C_*\) as the slice category of \(C\) over \(*\), i.e. as the following pullback square:
Objects of \(C_*\) are objects \(X\) of \(C\) equipped with a map \(x\colon * \to X\) from the terminal object, thought of as a chosen ‘basepoint’ of \(X\). We will frequently denote such objects by \((X,x)\), or sometimes abusively as \(X\).
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