Proposition 3.2 (0NJX) · ScalingStacks

Proposition 3.2. For any small stable $\infty$ -category ${\mathcal{A}}$ , the stable Yoneda embedding

{\mathcal{A}}\longrightarrow\mathrm{Fun}^{\ex}({\mathcal{A}}^{\op},{\mathcal{S}}_{\infty})

induces an equivalence $\Ind({\mathcal{A}})\simeq\mathrm{Fun}^{\ex}({\mathcal{A}}^{\op},{\mathcal{S}}_{\infty})$ .

0NJY

Proof. Clearly $\mathrm{Fun}^{\ex}({\mathcal{A}}^{\op},{\mathcal{S}}_{\infty})$ admits filtered colimits, as the filtered colimit of finite colimit preserving functors itself preserves finite colimits. This gives a map $\Ind({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}^{\ex}({\mathcal{A}}^{\op},{\mathcal{S}}_{\infty})$ which is evidently fully faithful since, using the fact that the usual Yoneda embedding is fully faithful and that mapping spaces between representables in $\mathrm{Fun}^{\ex}({\mathcal{A}}^{\op},{\mathcal{S}}_{\infty})$ are computed as the limit

\lim_{n}\Omega^{n}\map(a,\Sigma^{n}b)\simeq\lim_{n}\map(a,\Omega^{n}\Sigma^{n}b)\simeq\map(a,b).

To show that this map is also essentially surjective, we must show that any exact functor $f\colon{\mathcal{A}}^{\op}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}_{\infty}$ is ind-representable. Consider the pullback

where the right vertical map is the stable Yoneda embedding. We claim that the $\infty$ -category ${\mathcal{A}}_{/f}$ is filtered: to see this, let $K$ be a finite simplicial set and $K\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{A}}_{/f}$ a functor. Since both ${\mathcal{A}}$ and $\mathrm{Fun}^{\ex}({\mathcal{A}}^{\op},{\mathcal{S}}_{\infty})_{/f}$ admit finite colimits and both functors to $\mathrm{Fun}^{\ex}({\mathcal{A}}^{\op},{\mathcal{S}}_{\infty})$ preserve finite colimits, we may extend $K\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{A}}_{/f}$ to a colimit diagram $K^{\triangleright}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{A}}_{/f}$ . In particular, this gives a cone on $K\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{A}}_{/f}$ , which shows that ${\mathcal{A}}_{/f}$ is a filtered $\infty$ -category. Finally, since filtered colimits in $\mathrm{Fun}^{\ex}({\mathcal{A}}^{\op},{\mathcal{S}}_{\infty})$ are computed pointwise, it follows that $f$ is a colimit of the diagram ${\mathcal{A}}_{/f}\longrightarrow\mathrm{Fun}^{\ex}({\mathcal{A}}^{\op},{\mathcal{S}}_{\infty})$ , which is to say that it is ind-representable. ∎