Definition 5.4.1. An anima \(X\) is called \(0\)-connected if it is connected, i.e. if the set \(\pi _0(X)\) has a single element. It is called \(n\)-connected for \(n \geq 1\) if it is connected and the homotopy groups \(\pi _k(X)\) are trivial whenever \(k \leq n\). It is \((-1)\)-connected whenever it is non-empty. We write \[ \An _{\geq n} \subseteq \An \] for the full subcategory spanned by the \((n-1)\)-connected animae. Thus \(\An _{\geq 0}\) is the full subcategory of non-empty animae. Note that \(X\) is \(n\)-connected if and only if it is connected and \(\Omega X\) is \((n-1)\)-connected.
A spectrum \(X\) is called connective if, for every \(n \geq 1\), its \(n\)-th space \(X_n = \Omega ^{\infty - n}X\) is \((n-1)\)-connected. We write \[ \Sp _{\geq 0} \subseteq \Sp \] for the full subcategory spanned by the connective spectra. Connective spectra are also known as infinite loop spaces.
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