ScalingStacks

Definition 2.15. Let ${\mathcal{C}}$ be stable $\infty$-category. The spectral Yoneda embedding is the composite

$$\mathcal{C}\simeq\mathrm{Sp}(\mathcal{C}_{*})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Sp}(\mathrm{Fun}(\mathcal{C}^{\op},\mathrm{N}({\mathcal{T}})^{\cf})_{*})\simeq\mathrm{Fun}(\mathcal{C}^{\op},{\mathcal{S}}_{\infty}).$$

The mapping spectrum functor

$$\mathrm{Map}\colon\mathcal{C}^{\op}\times\mathcal{C}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}_{\infty}$$

is the adjoint of the spectral Yoneda embedding.