ScalingStacks

Given any $\infty$-category ${\mathcal{C}}$ with finite limits, we can form the stabilization $\Stab({\mathcal{C}})$ [53, §1.4]. The $\infty$-category $\Stab({\mathcal{C}})$ is stable and comes equipped with a limit-preserving functor

If in addition ${\mathcal{C}}$ is presentable, then $\Omega^{\infty}$ admits a left adjoint

by [53, 1.4.4.4].

We now recall an explicit model of the stabilization of an $\infty$-category in terms of spectrum objects [53, §1.4.2]. Recall that a spectrum object of a pointed $\infty$-category $\mathcal{C}$ consists of a functor $\mathrm{N}({\mathbb{Z}}\times{\mathbb{Z}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{C}$. In particular, there are families of objects $A(i,j)$ of $\mathcal{C}$ and maps $A(i,j)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A(i+1,j)$, $A(i,j)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}A(i,j+1)$ such that $A(i,j)$ is zero object whenever $i\neq j$ and the square

is cartesian for all $i$; consult [53, 1.4.2.4] for further details. Since the restriction of $A$ to the diagonal carries the nontrivial objects in $A$, we set $A_{i}=A(i,i)$ and often refer to $A$ simply by the collection of pointed objects $\{A_{i}\}$. We write $\mathrm{Sp}(\mathcal{C})$ for the $\infty$-category of spectrum objects in $\mathcal{C}$; $\mathrm{Sp}(\mathcal{C})$ comes equipped with a functor $\Omega^{\infty}\colon\mathrm{Sp}(\mathcal{C})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{C}$ which associates to the spectrum object $A$ its zero space $A_{0}=A(0,0)$. This is an explicit model for the stabilization $\Stab({\mathcal{C}})$ discussed previously. To ease notation, we will usually just write ${\mathcal{T}}_{\infty}\simeq\mathrm{N}(\mathcal{T}^{\cf})$ for the $\infty$-category of spaces and ${\mathcal{S}}_{\infty}\simeq\mathrm{Sp}({\mathcal{T}}_{\infty})\simeq\mathrm{N}({\mathcal{S}}^{\cf})$ for the $\infty$-category of spectra.

Now suppose that $\mathcal{C}$ is an arbitrary $\infty$-category. The Yoneda embedding $\mathcal{C}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}(\mathcal{C}^{\op},{\mathcal{T}}_{\infty})$ preserves finite limits (when they exist), so it induces a functor

on the level of spectrum objects, where $\mathcal{C}_{*}$ denotes the category of pointed objects in $\mathcal{C}$. Here the last equivalence follows from the fact that limits in functor categories are computed pointwise, and observe also that $\mathcal{C}_{*}$ will be empty unless $\mathcal{C}$ has a final object. On the other hand, if $\mathcal{C}$ is a stable $\infty$-category, then $\mathcal{C}\simeq\mathcal{C}_{*}$ and $\Omega^{\infty}\colon\mathrm{Sp}(\mathcal{C})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{C}$ is an equivalence with inverse $\Sigma^{\infty}\colon\mathcal{C}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Sp}(\mathcal{C})$ given by $(\Sigma^{\infty}a)_{i}=\Sigma^{i}a$ [53, 1.4.2.20]. This motivates the following definition:

Informally, the mapping spectrum is described by the formula

Note that this is a functor to the $\infty$-category of spectra; this is in contrast to the (point-set) mapping space functors from the category of quasicategories to the category of simplicial sets described in [52, 1.2.2] or [25].

We wish to characterize the image of ${\mathcal{C}}$ under the spectral Yoneda embedding:

When $\mathcal{C}$ is stable already, the following proposition gives an easy characterization of stably representable functors.