The basic modes

Theorem 18.5.3 (The basic modes, [Gepner et al. (2015), Sections 3--5]). The presentably symmetric monoidal \(\infty \)-categories \[ (\An _*,\wedge ), \qquad (\CMon (\An ),\otimes ), \qquad (\CGrp (\An ),\otimes ), \qquad (\Sp ,\otimes ) \] are modes. For every presentable \(\infty \)-category \(C\), there are natural equivalences \[ \begin {aligned} \An _*\otimes C &\simeq C_*, & \CMon (\An )\otimes C &\simeq \CMon (C), \\ \CGrp (\An )\otimes C &\simeq \CGrp (C), & \Sp \otimes C &\simeq \Sp (C). \end {aligned} \] Consequently, these four modes classify the properties recorded in Table 18.1. The functors in the sequence at the beginning of this section are morphisms of modes, meaning morphisms in \(\CAlg (\PrL )\).