ScalingStacks

Proof. The spectral Yoneda embedding

is adjoint to a simplicial functor

which evidently factors through the full simplicial subcategory $\Omega^{\infty}\Upsilon(\mathcal{C})$ spanned by the stably representable functors. The map $\mathcal{C}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{N}(\Omega^{\infty}\Upsilon(\mathcal{C}))$ is the adjoint of the resulting map $\mathfrak{C}[\mathcal{C}]\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Omega^{\infty}\Upsilon(\mathcal{C})$.

To see that this map is an equivalence, we observe first that it is essentially surjective: indeed, a stably representable cofibrant and fibrant functor $X\colon\mathfrak{C}[\mathcal{C}]^{\op}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}$ is necessarily of the form $X\simeq\mathrm{Map}(-,A)$ for some spectrum object $A=\{a_{i}\}$ of $\mathcal{C}\simeq\mathrm{N}(\mathfrak{C}[\mathcal{C}]^{\mathrm{fib}})$. Since $\mathcal{C}$ is stable, $a_{i}\simeq\Sigma^{i}a$ for some object $a$ of $\mathcal{C}$, so $X$ is in the image of $\mathcal{C}$ (which sends $a$ to the presheaf represented by $\Sigma^{\infty}a$). This map is also fully faithful, because if $a$ and $b$ are any pair of objects of $\mathcal{C}$, then

since $a\simeq\Omega^{\infty}\Sigma^{\infty}a$. ∎

Proof. Any stably representable functor $\mathcal{C}^{\op}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}_{\infty}$ is exact, giving the factorization

By proposition 3.2, we may rewrite this as $\mathcal{C}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ind(\mathcal{C})\simeq\mathrm{Fun}^{\ex}(\mathcal{C}^{\op},{\mathcal{S}}_{\infty})$ to see that, as an exact functor $\mathcal{C}^{\op}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}_{\infty}$, any stably representable functor is also compact. ∎

Proof. We first check that the construction of $\Upsilon$ induces a functor $\Cat^{\ex}_{\mathcal{T}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Cat_{\mathcal{S}}$. Let $f\colon{\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{D}}$ be a map of stable simplicial categories and write

for the induced spectral functor. Suppose that $X\colon\mathcal{C}^{\op}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}$ is projectively cofibrant and fibrant and that $\mathrm{N}(X)\colon\mathrm{N}(\mathcal{C})^{\op}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}_{\infty}$ is stably representable via the spectrum object $A=\{a_{i}\}$ in $\mathrm{N}\mathcal{C}$. Since the diagram

commutes (where the vertical maps are the stable Yoneda embeddings), we see that $f_{!}$ restricts to a spectral functor $\Upsilon({\mathcal{C}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Upsilon({\mathcal{D}})$.

To verify that $\Upsilon$ induces a simplicial functor, we must check that it preserves equivalences of stable simplicial categories. So suppose that $f\colon\mathcal{C}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{D}}$ is an equivalence of stable simplicial categories. Then it follows that $f_{!}^{\mathrm{cf}}$ is a DK-equivalence of spectral categories, as is its restriction to the stably representable objects. ∎

Proof. The first equivalence follows from the definition of $\Psi_{\tri}{\mathcal{A}}$ as the smallest stable subcategory of the stable $\infty$-category $\mathrm{N}(\mathrm{Fun}_{\mathcal{S}}({\mathcal{A}}^{\op},{\mathcal{S}})^{\mathrm{c}})[W^{-1}]$ containing the representables, together with the observations that $\mathrm{N}(\mathrm{Fun}_{\mathcal{S}}({\mathcal{A}}^{\op},{\mathcal{S}})^{\mathrm{c}})[W^{-1}]$ is compactly generated with compact objects

and $\Ind(\Psi_{\tri}{\mathcal{A}})\simeq\Ind(\Psi_{\perf}{\mathcal{A}})$ (as $\Ind$-categories are automatically idempotent-complete). The second equivalence follows immediately from proposition 3.2. ∎