ScalingStacks

We write ${\mathcal{T}}$ for the symmetric monoidal simplicial model category of simplicial sets and $\mathcal{S}$ for the symmetric monoidal simplicial model category of symmetric spectra [45]. Recall that a spectral category ${\mathcal{A}}$ is a category enriched in the category of symmetric spectra. Specifically, a spectral category is given by:

• •

  A class of objects $\mbox{obj}({\mathcal{A}})$,

• •

  for each pair of objects $(x,y)$ of ${\mathcal{A}}$, a symmetric spectrum ${\mathcal{A}}(x,y)$,

• •

  for each triple of objects $(x,y,z)$ of ${\mathcal{A}}$, a composition morphism in $\mathcal{S}$

  $${\mathcal{A}}(y,z)\wedge{\mathcal{A}}(x,y)\longrightarrow{\mathcal{A}}(x,z)\,,$$

  satisfying the usual associativity condition, and

• •

  for any object $x$ of ${\mathcal{A}}$, a morphism $\mathbb{S}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{A}}(x,x)$ in $\mathcal{S}$, satisfying the usual unit condition with respect to the above composition.

A spectral category is said to be small if its class of objects forms a set. We write $\Cat_{\mathcal{S}}$ the category of small spectral categories and spectral (enriched) functors. References on spectral categories are [10, §2], [69, Appendix A] and [74, §2].

We now briefly recall the Quillen model structure on spectral categories we work with in this paper. Given a spectral category ${\mathcal{A}}$, we can form a genuine category $[{\mathcal{A}}]$ by keeping the same set of objects and defining the set of morphisms between $x$ and $y$ in $[{\mathcal{A}}]$ to be the set of morphisms in the homotopy category $\Ho(\mathcal{S})$ from the sphere spectrum $\mathbb{S}$ to ${\mathcal{A}}(x,y)$. We obtain in this way a functor

with values in the category of small categories. Equivalently, we can think of $[-]$ as computed by passing to $\pi_{0}$ on the morphism spectra, and so we will also refer to $[{\mathcal{A}}]$ as the homotopy category $\Ho({\mathcal{A}})$.

Recall from [74, § 2] the natural adjunction

between spectral and simplicial categories, where $\Omega^{\infty}$ (also denoted $(-)_{0}$) is the space of maps from the unit (equivalently, restriction to the $0$-th space of the spectrum). We will use this adjunction to pass between spectral categories and $\infty$-categories. Using the model structure on simplicial categories of [6], the pair $(\Sigma^{\infty}_{+},\Omega^{\infty})$ is a Quillen adjunction.

For technical control, we require the following corollary which sharpens the description of the model structure, providing a combinatorial model category. (For references for Jeff Smith’s theory of combinatorial model categories, see [2] or [24].)

Let ${\mathcal{A}}$ be a (fixed) small spectral category and let ${\mathcal{A}}^{\op}$ denote the opposite spectral category, defined by ${\mathcal{A}}^{\op}(x,y)={\mathcal{A}}(y,x)$.

By [69, A.1.1], $\widehat{{\mathcal{A}}}$ can be given a combinatorial spectral model structure in which the weak equivalences are the pointwise stable equivalences and the fibrations are pointwise fibrations (referred to as the projective model structure). We will denote by $\widehat{{\mathcal{A}}}^{\cf}$ the full spectral subcategory of $\widehat{{\mathcal{A}}}$ on the cofibrant and fibrant ${\mathcal{A}}$-modules, and by ${\mathcal{D}}({\mathcal{A}})$ the derived category of ${\mathcal{A}}$, i.e., the homotopy category $\Ho(\widehat{{\mathcal{A}}})$ associated to the model structure. As usual, there is an equivalence $[\widehat{{\mathcal{A}}}^{\cf}]\simeq{\mathcal{D}}({\mathcal{A}})$.

Notice that we have a (fully faithful) spectral Yoneda embedding ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widehat{{\mathcal{A}}}$ which sends the object $z$ to the functor ${\mathcal{A}}(-,z)\colon{\mathcal{A}}^{\op}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{S}$ represented by $z$. Note that when ${\mathcal{A}}$ is fibrant, the Yoneda embedding lands in $\widehat{{\mathcal{A}}}^{\cf}$. By [69, §A.1], a spectral functor $F\colon{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ gives rise to a restriction/extension Quillen adjunction

and therefore a total left-derived functor $\mathbb{L}F_{!}\colon{\mathcal{D}}({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{D}}({\mathcal{B}})$.

We will be most interested in spectral categories ${\mathcal{A}}$ for which the homotopy category $\Ho({\mathcal{A}})$ has a triangulated structure compatible with the mapping spectra; we refer to [10, 4.4] for the definition of a pretriangulated spectral category, and highlight the essential consequence [10, 4.6] that the homotopy category of a pretriangulated spectral category is triangulated (with distinguished triangles given by the Puppe sequences). This is the stable homotopy theory analogue of the notion of a pretriangulated dg-category. A spectral functor between pretriangulated spectral categories is a DK-equivalence if and only if it induces an equivalent on homotopy categories [11, 5.7]. Using the Yoneda embedding, we can construct minimal pretriangulated categories containing the spectral category ${\mathcal{A}}$.

Given a spectral category ${\mathcal{A}}$, the proof of [10, 4.5] constructs a functorial “triangulated closure” $\widehat{{\mathcal{A}}}_{\tri}$ which is a pretriangulated spectral category. Briefly, $\widehat{{\mathcal{A}}}_{\tri}$ consists of the subcategory of cofibrant-fibrant objects in $\widehat{{\mathcal{A}}}$ which have the homotopy type of finite cell objects (in the projective model structure). Using retracts of finite cell objects instead [10, 4.5] produces a functorial “thick closure” $\widehat{{\mathcal{A}}}_{\perf}$, which is an idempotent-complete pretriangulated spectral category.

We start with a spectral functor $F\colon{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$, and tacitly assume we have performed a functorial fibrant replacement. We denote by $F_{!}^{\cf}\colon\widehat{{\mathcal{A}}}^{\cf}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widehat{{\mathcal{B}}}^{\cf}$ the composite of $F_{!}$ with a fibrant replacement (note that we do not need a cofibrant replacement here since $F_{!}$ preserves cofibrant objects) to obtain

Since this is a model of the derived functor of $F_{!}$ as a left Quillen functor, it preserves homotopy colimits and thus sends modules of the homotopy type of finite cell ${\mathcal{A}}$-modules to modules of the homotopy type of finite cell ${\mathcal{B}}$-modules and hence perfect ${\mathcal{A}}$-modules to perfect ${\mathcal{B}}$-modules. Therefore, the following definitions make sense.

We can relate these notions to definitions purely on the level of triangulated categories (the relationship between triangulated constructions and enriched constructions is discussed further in Section 5). For a spectral category ${\mathcal{A}}$, let ${\mathcal{D}}_{\tri}({\mathcal{A}})$ denote the smallest triangulated subcategory of ${\mathcal{D}}({\mathcal{A}})$ containing the image of the ${\mathcal{A}}$ under the Yoneda embedding, and ${\mathcal{D}}_{\perf}({\mathcal{A}})$ denote the smallest thick subcategory of ${\mathcal{D}}({\mathcal{A}})$ containing the image of ${\mathcal{A}}$ under the Yoneda embedding. Observe that ${\mathcal{D}}_{\tri}({\mathcal{A}})\simeq\Ho(\widehat{{\mathcal{A}}}_{\tri})$ and ${\mathcal{D}}_{\perf}({\mathcal{A}})\simeq\Ho(\widehat{{\mathcal{A}}}_{\perf})$. As a consequence, we obtain the following proposition.

Finally, note that we can use $\Omega^{\infty}$ to obtain simplicial models of the triangulated and thick closures. Define $\Mod({\mathcal{A}})$ to be the simplicial category $\Omega^{\infty}\widehat{{\mathcal{A}}}^{\cf}$, $\Mod({\mathcal{A}})_{\tri}$ to be the simplicial category $\Omega^{\infty}\widehat{{\mathcal{A}}}_{\tri}$, and $\Mod({\mathcal{A}})_{\perf}$ to be the simplicial category $\Omega^{\infty}\widehat{{\mathcal{A}}}_{\perf}$. Of course, it is also possible to give intrinsic definitions of the latter two categories in terms of $\Mod({\mathcal{A}})$.

Summarizing the relationships between the various categories, we have the following commutative diagram (with horizontal arrows induced by the Yoneda embedding and subsequent inclusions):