Metrics on triangulated categories

TL;DR

This survey explores metrics on triangulated categories, showing that their completions contain interesting triangulated subcategories, with applications to derived categories of coherent sheaves and perfect complexes.

Contribution

It specializes general metric constructions to triangulated categories and proves that completions contain triangulated subcategories, revealing new structural insights.

Findings

Completion of a triangulated category with a good metric contains a triangulated subcategory.

Special cases include derived categories of coherent sheaves and perfect complexes.

Provides new examples of triangulated categories arising from metric completions.

Abstract

In this survey we explain the results of the recent article arXiv:1806.06471. Following a 1973 article by Lawvere one can define metrics on categories, and following Kelly's 1982 book one can complete a category with respect to its metric. We specialize these general constructions to triangulated categories, and restrict our attention to "good metrics". And the remarkable new theorem is that, when we start with a triangulated category $\mathcal S$ with a good metric, its completion $\mathfrak{L}(\mathcal{S})$ contains an interesting subcategory $\mathfrak{S}(\mathcal{S})$ which is always triangulated. As special cases we obtain $\mathcal{H}^0(\mathrm{Perf}(X))$ and $D^b_{\mathrm{coh}}(X)$ from each other. We also give a couple of other examples.

Equations

$$\text{Length}(gf)\le \text{Length}(f)+\text{Length}(g).$$

$$\begin{array}[]{ccc}\{\text{\rm Length}_{1}(a\rightarrow b)<\delta\}&\quad\Longrightarrow&\{\text{\rm Length}_{2}(a\rightarrow b)<\varepsilon\}\ ,\\ \{\text{\rm Length}_{2}(a\rightarrow b)<\delta\}&\quad\Longrightarrow&\{\text{\rm Length}_{1}(a\rightarrow b)<\varepsilon\}\ .\end{array}$$

$$\text{Length}(E_i\to E_j)<\epsilon$$

$$\{\text{Length}(a\to b)<\epsilon \}⟹\{F(b)⟶F(a)\text{ is an isomorphism}\}.$$

$$\mathcal{C}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$a\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\text{Length}(f)=\text{Length}(g).$$

$$a\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\text{Length}(f)=\text{Length}(g).$$

$$B_n=\left\{x\in \mathcal{S}|\text{the morphism }0⟶x\text{ has length }\le \frac{1}{n}\right\}.$$

$$\text{Length}(f)=\inf \left\{\frac{1}{n}|z\in B_n\right\}.$$

$$\coprod _{i=1}^{\infty }{T}_i\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$T_n\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\tilde{Y}(E_n)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\hat{\mathfrak{S}}(\mathcal{S})\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\hat{\mathfrak{L}}(\mathcal{S})\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\widetilde{Y}^{-1}\big{[}\mathfrak{S}({\mathscr{S}})\big{]}\quad=\quad F^{-1}{\mathscr{Y}}^{-1}_{\widehat{\mathfrak{L}}({\mathscr{S}})}\big{[}\mathfrak{S}({\mathscr{S}})\big{]}\quad=\quad F^{-1}\big{[}\widehat{\mathfrak{S}}({\mathscr{S}})\big{]}$$

$$H(X,n)={\mathrm{colim}}_{⟶}{\mathrm{Hom}}_{\text{CW-complexes}}({\mathbb{S}}^k,{\Sigma }^{n+k}X),$$

$$\widehat{\mathfrak{S}}({\mathscr{S}})\quad=\quad\left\{x\in{\mathscr{T}}\left|\begin{array}[]{c}H^{i}(x)=0\text{ for all but finitely many }i\in{\mathbb{Z}},\text{ and}\\ H^{i}(x)\text{ is a finitely generated ${\mathbb{Z}}$--module for all }i\in{\mathbb{Z}}\end{array}\right.\right\}\ .$$

$$\pi_{-i}\big{[}K({\mathbb{Z}},0)\big{]}\quad=\quad H^{i}\big{[}K({\mathbb{Z}},0)\big{]}\quad=\quad\left\{\begin{array}[]{lll}{\mathbb{Z}}&&\text{if }i=0\\ 0&&\text{otherwise}\end{array}\right.$$

$$\begin{array}[]{cll}{\mathop{\rm Hom}}(\mathbb{S},\Sigma^{n}\mathbb{S})=0&&\text{if }0<n\\ {\mathop{\rm Hom}}\big{(}K({\mathbb{Z}},0),\Sigma^{n}K({\mathbb{Z}},0)\big{)}=0&&\text{if }n<0\\ \end{array}$$

$$\begin{array}[]{cll}{\mathop{\rm Hom}}\big{(}K({\mathbb{Z}},0),\Sigma^{n}\mathbb{S}\big{)}=0&&\text{for all }n\neq 1\end{array}$$

$$\begin{array}[]{cclcll}\text{For any pair }G,H\in{\mathscr{S}}&&{\mathop{\rm Hom}}(G,\Sigma^{n}H)=0&&\text{ if }&0\ll n\\ \text{For any pair }G,H\in\widehat{\mathfrak{S}}({\mathscr{S}})&&{\mathop{\rm Hom}}\big{(}G,\Sigma^{n}H\big{)}=0&&\text{ if }&n\ll 0\\ \text{For any pair }G\in\widehat{\mathfrak{S}}({\mathscr{S}}),\,H\in{\mathscr{S}}&&{\mathop{\rm Hom}}\big{(}G,\Sigma^{n}H\big{)}=0&&\text{ if }&|n|\gg 0\end{array}$$

$$\begin{array}[]{cclcll}\text{For any pair }G,H\in{\mathscr{S}}&&{\mathop{\rm Hom}}(G,\Sigma^{n}H)&&\text{ is a f.g.~{}${\mathbb{Z}}$--module}\\ \text{For any pair }G,H\in\widehat{\mathfrak{S}}({\mathscr{S}})&&{\mathop{\rm Hom}}\big{(}G,\Sigma^{n}H\big{)}&&\text{ is a f.g.~{}${\mathbb{Z}}$--module}\\ \text{For any pair }G\in\widehat{\mathfrak{S}}({\mathscr{S}}),\,H\in{\mathscr{S}}&&{\mathop{\rm Hom}}\big{(}G,\Sigma^{n}H\big{)}&&\text{ is a ${\mathbb{Q}}$--vector space}\end{array}$$

$${\mathbf{D}}_{\text{\rm sing}}(R)\quad=\quad\frac{{\mathbf{D}}^{b}(\mathop{R\textup{--mod}})}{{\mathscr{H}}^{0}\big{(}\text{\bf Perf}(R)\big{)}}\ ,$$

Full text

This turns out to be particularly interesting when the category is triangulated and the Lawvere metric is good; a metric is good if it is translation invariant and the balls of radius $\varepsilon>0$ shrink rapidly enough as $\varepsilon$ decreases. The definitions are all made precise at the beginning of the note. And the main theorem is that a certain natural subcategory $\mathfrak{S}({\mathscr{S}})$, of the completion of ${\mathscr{S}}$ with respect to a good metric, is triangulated.

There is also a theorem which, under restrictive conditions, gives a procedure for computing $\mathfrak{S}({\mathscr{S}})$. As examples we discuss the special cases (1) where ${\mathscr{S}}$ is the homotopy category of finite spectra, and (2) where ${\mathscr{S}}={\mathbf{D}}^{b}(\mathop{R\textup{--mod}})$, the derived category of bounded complexes of finitely generated $R$–modules over a noetherian ring $R$.

The research was partly supported by the Australian Research Council

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Lawvere’s article does many other things and has had an enormous influence over the years—when I last checked on Google Scholar it had 732 citations. To the best of my knowledge the myriad applications have essentially all gone in directions totally different from the one we will be pursuing in this note. There is only a handful of exceptions, we will cite these when they become relevant. **

We begin with a string of definitions.

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whenever $i,j>M$ and $i\leq j$.

We are accustomed from analysis to the idea of completing a metric space with respect to its metric, and now we want to do the same for Lawvere metrics on categories. And the idea is simple enough: the Yoneda embedding takes any category ${\mathscr{C}}$ to a subcategory of a cocomplete category, with the traditional definition that a category is cocomplete if all small colimits exist. Hence the completion of ${\mathscr{C}}$ should just be the closure of the image under Yoneda of ${\mathscr{C}}$. We make this precise in:

1. (i)

Let $\mathfrak{L}^{\prime}({\mathscr{C}})$ be the completion of ${\mathscr{C}}$, meaning the full subcategory of ${\mathop{\rm Hom}}[{\mathscr{C}}^{\hbox{\rm\tiny op}},\text{\rm Set}]$ whose objects are the colimits in ${\mathop{\rm Hom}}[{\mathscr{C}}^{\hbox{\rm\tiny op}},\text{\rm Set}]$ of Cauchy sequences in ${\mathscr{C}}$. 2. (ii)

Let $\mathfrak{C}^{\prime}({\mathscr{C}})$ be the full subcategory of ${\mathop{\rm Hom}}[{\mathscr{C}}^{\hbox{\rm\tiny op}},\text{\rm Set}]$ whose objects we will call compactly supported. An object $F\in{\mathop{\rm Hom}}[{\mathscr{C}}^{\hbox{\rm\tiny op}},\text{\rm Set}]$, that is a functor $F:{\mathscr{C}}^{\hbox{\rm\tiny op}}\longrightarrow\text{\rm Set}$, is declared to be compactly supported if it takes sufficiently short morphisms to isomorphisms. That is: $F$ belongs to $\mathfrak{C}^{\prime}({\mathscr{C}})$ if there exists an $\varepsilon>0$ such that

[TABLE] 3. (iii)

Now let $\mathfrak{S}^{\prime}({\mathscr{C}})=\mathfrak{C}^{\prime}({\mathscr{C}})\cap\mathfrak{L}^{\prime}({\mathscr{C}})$.

Next assume the category ${\mathscr{C}}$ is pre-additive. This means that ${\mathop{\rm Hom}}(a,b)$ is an abelian group for every pair of objects $a,b\in{\mathscr{C}}$, and the composition is bilinear.222In the more recent literature what used to be called pre-additive categories goes by the name ${\mathbb{Z}}$–linear categories, or even just ${\mathbb{Z}}$–categories. In this situation the Yoneda map factors as a composite

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where $\text{\rm Mod--}{\mathscr{C}}$ is the category whose objects are the additive functors of the form ${\mathscr{C}}^{\hbox{\rm\tiny op}}\longrightarrow\mathrm{Ab}$. And since $\text{\rm Mod--}{\mathscr{C}}$ is cocomplete we now have the option of taking the closure of the image of $\widetilde{Y}$ instead of the closure of the image of $Y$. This leads us to

1. (i)

Let $\mathfrak{L}({\mathscr{C}})$ be the completion of $\widetilde{Y}({\mathscr{C}})$ in $\text{\rm Mod--}{\mathscr{C}}$; it is the full subcategory of $\text{\rm Mod--}{\mathscr{C}}$ whose objects are the colimits in $\text{\rm Mod--}{\mathscr{C}}$ of Cauchy sequences in ${\mathscr{C}}$. 2. (ii)

Let $\mathfrak{C}({\mathscr{C}})=\Phi^{-1}\mathfrak{C}^{\prime}({\mathscr{C}})$. 3. (iii)

Finally let $\mathfrak{S}({\mathscr{C}})=\mathfrak{C}({\mathscr{C}})\cap\mathfrak{L}({\mathscr{C}})$.

Categories with metrics—that is what Lawvere [Law73] calls normed categories—may be viewed as categories enriched over a certain closed monoidal category, see Betti and Galuzzi [BG75] for a detailed exposition. And there is a notion of completing an enriched category with respect to a class of colimits, the reader can find it already in Kelly’s book [Kel82], but for a more direct approach see Kelly and Schmitt [KS05]. It doesn’t seem automatic that the specialization of the general theory to the case at hand agrees with what we’ve done in this article. That is: using Kelly’s construction we may complete a normed category ${\mathscr{C}}$, enriched as in [BG75], with respect to Cauchy sequences—and what’s obtained doesn’t in general seem to agree with our $\mathfrak{L}^{\prime}({\mathscr{C}})$. There are conditions that suffice to guarantee agreement: in Kubiś [Kub] the reader can see that adding the extra axiom that $\text{\rm Length}(f)\leq\text{\rm Length}(gf)+\text{\rm Length}(g)$ is sufficient. In this article we work mostly with “good metrics”, which will be spelled out in Definition 10. Our good metrics happen not to satisfy the Kubiś axiom. The interested reader can nevertheless check that restricting to good metrics also suffices to guarantee the agreement of $\mathfrak{L}^{\prime}({\mathscr{C}})$ with the Cauchy completion due to Kelly.

For yet another construction of the category $\mathfrak{L}^{\prime}({\mathscr{C}})$ see Krause [Kra]. Krause only looks at one particular metric but the method generalizes. In Krause’s approach the category $\mathfrak{L}^{\prime}({\mathscr{C}})$ is presented as the Gabriel-Zisman localization (see [GZ67]) of the category of Cauchy sequences, where one formally inverts the Ind-isomorphisms.

The idea of studying the categories $\mathfrak{C}^{\prime}({\mathscr{C}})$ and $\mathfrak{S}^{\prime}({\mathscr{C}})$ seems to have arisen only in [Neeb]. **

In the interest of full disclosure: in the generality of the paragraphs above I can’t prove anything worthwhile. The only obvious observation is that the constructions are robust under replacing one metric by an equivalent other. The Cauchy sequences depend only on the equivalence class of the metric, hence so do the categories $\mathfrak{L}^{\prime}({\mathscr{C}})$ and $\mathfrak{L}({\mathscr{C}})$. The definitions of the categories $\mathfrak{C}^{\prime}({\mathscr{C}})$ and $\mathfrak{C}({\mathscr{C}})$ make it clear that these two categories are also unperturbed by replacing a metric by an equivalent. Hence the same is true for $\mathfrak{S}^{\prime}({\mathscr{C}})=\mathfrak{C}^{\prime}({\mathscr{C}})\cap\mathfrak{L}^{\prime}({\mathscr{C}})$ and $\mathfrak{S}({\mathscr{C}})=\mathfrak{C}({\mathscr{C}})\cap\mathfrak{L}({\mathscr{C}})$.

So much for triviality. To get anywhere we need to narrow our attention considerably. **

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we postulate that

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Given any morphism $f:a\longrightarrow b$ we may form the homotopy cartesian square

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and our assumption tells us that

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Hence it suffices to know the lengths of the morphisms $0\longrightarrow x$. Replacing the metric by an equivalent, if necessary, we may assume our metric takes values in the set of rational numbers of the form $\left\{\frac{1}{n}\,\,\left|\,\,n\in{\mathbb{N}}\right\}\right.$. To know everything about the metric it therefore suffices to specify the balls

[TABLE]

To paraphrase the discussion above: if $f:x\longrightarrow y$ is a morphism, to compute its length you complete to a triangle $x\stackrel{{\scriptstyle f}}{{\longrightarrow}}y\longrightarrow z$, and then

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Furthermore we will restrict our attention to non-archimedian metrics, that is metrics that satisfy the strong triangle inequality. This means: if $x\stackrel{{\scriptstyle f}}{{\longrightarrow}}y\stackrel{{\scriptstyle g}}{{\longrightarrow}}z$ are composable morphisms, then \text{Length}(gf)\leq\max\big{(}\text{Length}(f),\text{Length}(g)\big{)}. By the translation-invariance it suffices to consider the case $x=0$; that is it suffices to show that the composable morphisms $0\stackrel{{\scriptstyle f}}{{\longrightarrow}}y\stackrel{{\scriptstyle g}}{{\longrightarrow}}z$ satisfy \text{Length}(gf)\leq\max\big{(}\text{Length}(f),\text{Length}(g)\big{)}. Completing $g$ to a triangle $y\stackrel{{\scriptstyle g}}{{\longrightarrow}}z\longrightarrow w$ this comes down to $\{y,w\in B_{n}\}\Longrightarrow\{z\in B_{n}\}$. **

The discussion above motivates

1. (i)

$B_{n}*B_{n}=B_{n}$, which means that if there exists a triangle $b\longrightarrow x\longrightarrow b^{\prime}$ with $b,b^{\prime}\in B_{n}$, then $x\in B_{n}$. 2. (ii)

$\Sigma^{-1}B_{n+1}\cup B_{n+1}\cup\Sigma B_{n+1}\subset B_{n}$.

Note that we are not assuming that the metric is compatible with any other automorphism of ${\mathscr{S}}$. **

1. (i)

B_{n}=\{s\in{\mathscr{S}}\mid H^{i}(s)=0\text{ for all i in the range }i>-n\}. 2. (ii)

B_{n}=\{s\in{\mathscr{S}}\mid H^{i}(s)=0\text{ for all i in the range }i<n\}. 3. (iii)

B_{n}=\{s\in{\mathscr{S}}\mid H^{i}(s)=0\text{ for all i in the range }-n<i<n\}

Note that if $\{B_{n},\,n\in{\mathbb{N}}\}$ define a good metric on ${\mathscr{S}}$ then $\{B^{\hbox{\rm\tiny op}}_{n},\,n\in{\mathbb{N}}\}$ define a good metric on ${\mathscr{S}}^{\hbox{\rm\tiny op}}$, which we will call the dual metric. Now a homological functor $H:{\mathscr{S}}\longrightarrow{\mathscr{A}}$ has a dual $H^{\hbox{\rm\tiny op}}:{\mathscr{S}}^{\hbox{\rm\tiny op}}\longrightarrow{\mathscr{A}}^{\hbox{\rm\tiny op}}$, and the reader can check that (i), applied to $H^{\hbox{\rm\tiny op}}:{\mathscr{S}}^{\hbox{\rm\tiny op}}\longrightarrow{\mathscr{A}}^{\hbox{\rm\tiny op}}$, gives a good metric equal to the dual of that obtained from (ii) applied to $H:{\mathscr{S}}\longrightarrow{\mathscr{A}}$.

The metric of (iii) is self-dual. **

One more definition before the first theorem.

And now we come to

The proof of a slightly stronger theorem [the hypotheses on the metric are slightly less restrictive] may be found in [Neeb, Theorem 2.11].

Then, out of all the seemingly pointless formalism, we magically pulled out Theorem 15. Perhaps it takes an expert to appreciate how surprising the result is. Triangulated categories have been around since the early 1960s—meaning for about 55 years. And the conventional wisdom has always been that they don’t reproduce. Until very recently there were no interesting recipes that began with a triangulated category ${\mathscr{S}}$, and out of it cooked up another triangulated category ${\mathscr{T}}$—in this context we view (full) triangulated subcategories and Verdier quotients as dull, trivial constructions. In more detail: the first whiff of such a recipe came in 2005 in Keller [Kel05]. Keller proved that, given a triangulated category ${\mathscr{S}}$ and an automorphism $\sigma:{\mathscr{S}}\longrightarrow{\mathscr{S}}$, then the category ${\mathscr{T}}={\mathscr{S}}/\sigma$ sometimes [rarely] has a triangulated structure so that the quotient map is triangulated—but the conditions are very stringent. And the only other known recipe was found in 2011 by Balmer [Bal11]: given a separable monoid $R$ in a tensor triangulated category ${\mathscr{S}}$, the category ${\mathscr{T}}$ whose objects are the $R$–modules in ${\mathscr{S}}$, and whose morphisms are those morphisms in ${\mathscr{S}}$ which respect the $R$–module structure, is triangulated. The distinguished triangles in ${\mathscr{T}}$ are precisely those sequences $T\longrightarrow T^{\prime}\longrightarrow T^{\prime\prime}\longrightarrow\Sigma T$ whose image in ${\mathscr{S}}$ is a distinguished triangle. OK: these two relatively recent exceptions aside, the accepted wisdom has long been that you need some enhancement to produce triangulated categories in a nontrivial way.

Theorem 15 gives a third recipe, and the natural question is whether the end product is of any value. Given an input triangulated category ${\mathscr{S}}$, together with its good metric, is the output category $\mathfrak{S}({\mathscr{S}})$ a worthwhile object of study? And to answer this we need examples. In Example 12 we saw three ways to produce good metrics, out of any homological functor $H:{\mathscr{S}}\longrightarrow{\mathscr{A}}$. For each of these the question arises: what is the triangulated category $\mathfrak{S}({\mathscr{S}})$? Is it of any interest?

In general I don’t know how to compute $\mathfrak{S}({\mathscr{S}})$. The only procedure I know so far assumes that ${\mathscr{S}}$ has an embedding into a larger triangulated category ${\mathscr{T}}$, and this embedding satisfies a strong condition. In the presence of such an embedding $\mathfrak{S}({\mathscr{S}})$ may be computed as a triangulated subcategory of ${\mathscr{T}}$. Below we will spell out carefully the exact statements. **

In order to make the above precise we need some more definitions—my apologies to the reader, we will get to the point in Theorem 20.

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where $\text{\rm(shift)}:\coprod_{i=1}^{\infty}T_{i}\longrightarrow\coprod_{i=1}^{\infty}T_{i}$ is the unique map rendering commutative, for any integer $n\geq 1$, the square below

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with $(\text{\rm inc})$ being the canonical inclusion into the coproduct. The object \begin{picture}(40.0,10.0)\put(20.0,0.0){\makebox(0.0,0.0)[b]{\text{\rm Hocolim}}} \put(5.0,-2.0){\vector(1,0){30.0}} \end{picture}\,\,T_{*} is only defined up to (non-canonical) isomorphism in ${\mathscr{T}}$, but the isomorphism can be assumed to respect the map $\varphi$. Hence any such isomorphism, between two candidates for \begin{picture}(40.0,10.0)\put(20.0,0.0){\makebox(0.0,0.0)[b]{\text{\rm Hocolim}}} \put(5.0,-2.0){\vector(1,0){30.0}} \end{picture}\,\,T_{*}, will respect all the composites T_{n}\stackrel{{\scriptstyle\text{\rm inc}}}{{\longrightarrow}}\coprod_{i=1}^{\infty}T_{i}\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}\begin{picture}(40.0,10.0)\put(20.0,0.0){\makebox(0.0,0.0)[b]{\text{\rm Hocolim}}} \put(5.0,-2.0){\vector(1,0){30.0}} \end{picture}\,\,T_{*}; we write these as \varphi_{n}:T_{n}\longrightarrow\begin{picture}(40.0,10.0)\put(20.0,0.0){\makebox(0.0,0.0)[b]{\text{\rm Hocolim}}} \put(5.0,-2.0){\vector(1,0){30.0}} \end{picture}\,\,T_{*}. The vanishing of the composite $\varphi\circ(\mathrm{id}-\text{\rm shift})$, in the displayed maps of the triangle above, guarantees that, for each integer $n\geq 1$, the composite T_{n}\longrightarrow T_{n+1}\stackrel{{\scriptstyle\varphi_{n+1}}}{{\longrightarrow}}\begin{picture}(40.0,10.0)\put(20.0,0.0){\makebox(0.0,0.0)[b]{\text{\rm Hocolim}}} \put(5.0,-2.0){\vector(1,0){30.0}} \end{picture}\,\,T_{*} must be equal to \varphi_{n}:T_{n}\longrightarrow\begin{picture}(40.0,10.0)\put(20.0,0.0){\makebox(0.0,0.0)[b]{\text{\rm Hocolim}}} \put(5.0,-2.0){\vector(1,0){30.0}} \end{picture}\,\,T_{*}.

If we are given a sequence $E_{1}\longrightarrow E_{2}\longrightarrow E_{3}\longrightarrow\cdots$ in the category ${\mathscr{S}}$, then the functor $F$ takes it to a sequence $F(E_{1})\longrightarrow F(E_{2})\longrightarrow F(E_{3})\longrightarrow\cdots$ in the category ${\mathscr{T}}$. The paragraph above gives, for each $n$, a map \varphi_{n}:F(E_{n})\longrightarrow\begin{picture}(40.0,10.0)\put(20.0,0.0){\makebox(0.0,0.0)[b]{\text{\rm Hocolim}}} \put(5.0,-2.0){\vector(1,0){30.0}} \end{picture}\,\,F(E_{*}). Applying to this the functor ${\mathscr{Y}}$ we deduce the second morphism in the composable pair below

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where the first morphism comes from the canonical isomorphism $\widetilde{Y}\cong{\mathscr{Y}}F$. And these maps assemble to a single morphism {\displaystyle\mathop{\rm colim}_{\displaystyle\longrightarrow}}\,\widetilde{Y}(E_{*})\longrightarrow{\mathscr{Y}}\big{(}\begin{picture}(40.0,10.0)\put(20.0,0.0){\makebox(0.0,0.0)[b]{\text{\rm Hocolim}}} \put(5.0,-2.0){\vector(1,0){30.0}} \end{picture}\,\,F(E_{*})\big{)}, unique up to (non-canonical) isomorphism. Hence postulating that this map is an isomorphism, as in Definition 17, makes sense independent of choices.

Note that this is a strong restriction. If ${\mathscr{S}}$ has countable coproducts we might be tempted to let $F$ be the identity $\mathrm{id}:{\mathscr{S}}\longrightarrow{\mathscr{S}}$. But then it becomes a strong hypothesis to assume that the Cauchy sequences all satisfy the condition that {\displaystyle\mathop{\rm colim}_{\displaystyle\longrightarrow}}\,\widetilde{Y}(E_{*})\longrightarrow\widetilde{Y}\big{(}\begin{picture}(40.0,10.0)\put(20.0,0.0){\makebox(0.0,0.0)[b]{\text{\rm Hocolim}}} \put(5.0,-2.0){\vector(1,0){30.0}} \end{picture}\,\,E_{*}\big{)} is an isomorphism. **

1. (i)

The full subcategory $\widehat{\mathfrak{L}}({\mathscr{S}})\subset{\mathscr{T}}$ has for objects all the homotopy colimits of Cauchy sequences in ${\mathscr{S}}$. 2. (ii)

The full subcategory $\widehat{\mathfrak{S}}({\mathscr{S}})\subset{\mathscr{T}}$ is given by the formula \widehat{\mathfrak{S}}({\mathscr{S}})=\widehat{\mathfrak{L}}({\mathscr{S}})\cap{\mathscr{Y}}^{-1}\big{(}\mathfrak{C}({\mathscr{S}})\big{)}, with $\mathfrak{C}({\mathscr{S}})\subset\text{\rm Mod--}{\mathscr{S}}$ as in Definition 5(ii).

The proof of a slightly stronger theorem [once again, the hypotheses on the metric are slightly less restrictive] may be found in [Neeb, Theorem 3.15].

The functor $F:{\mathscr{S}}\longrightarrow{\mathscr{T}}$ is a fully faithful, triangulated functor, while the subcategory $\widehat{\mathfrak{S}}({\mathscr{S}})\subset{\mathscr{T}}$ is triangulated. Hence F^{-1}\big{[}\widehat{\mathfrak{S}}({\mathscr{S}})\big{]} is a triangulated subcategory of ${\mathscr{S}}$, and the functor $F$ restricts to a fully faithful, triangulated functor F^{-1}\big{[}\widehat{\mathfrak{S}}({\mathscr{S}})\big{]}\longrightarrow\widehat{\mathfrak{S}}({\mathscr{S}}).

Now $\widehat{\mathfrak{S}}({\mathscr{S}})$ lies in $\widehat{\mathfrak{L}}({\mathscr{S}})\subset{\mathscr{T}}$, and the functor ${\mathscr{Y}}:{\mathscr{T}}\longrightarrow\text{\rm Mod--}{\mathscr{S}}$ obviously takes $\widehat{\mathfrak{L}}({\mathscr{S}})\subset{\mathscr{T}}$ to $\mathfrak{L}({\mathscr{S}})\subset\text{\rm Mod--}{\mathscr{S}}$; hence ${\mathscr{Y}}$ restricts to a functor ${\mathscr{Y}}|_{\widehat{\mathfrak{L}}({\mathscr{S}})}:\widehat{\mathfrak{L}}({\mathscr{S}})\longrightarrow\mathfrak{L}({\mathscr{S}})$. And it turns out to be easy to show that the functor ${\mathscr{Y}}|_{\widehat{\mathfrak{L}}({\mathscr{S}})}$ is essentially surjective, full and conservative. This means: every object in $\mathfrak{L}({\mathscr{S}})$ is isomorphic to an object in the image of ${\mathscr{Y}}|_{\widehat{\mathfrak{L}}({\mathscr{S}})}$, any morphism in $\mathfrak{L}({\mathscr{S}})$ between objects in the image of ${\mathscr{Y}}|_{\widehat{\mathfrak{L}}({\mathscr{S}})}$ is in the image of ${\mathscr{Y}}|_{\widehat{\mathfrak{L}}({\mathscr{S}})}$, and a morphism in $\widehat{\mathfrak{L}}({\mathscr{S}})$ is an isomorphism if and only if ${\mathscr{Y}}|_{\widehat{\mathfrak{L}}({\mathscr{S}})}$ takes it to an isomorphism. And the relevance of this for us is that the commutative square

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is a strict pullback square. The point is that, from their definitions, the categories $\widehat{\mathfrak{S}}({\mathscr{S}})$ and $\mathfrak{S}({\mathscr{S}})$ are replete subcategories in, respectively, ${\mathscr{T}}$ and $\text{\rm Mod--}{\mathscr{S}}$; this means they contain all isomorphs of any of their objects. Theorem 20 tells us that the vertical map on the left is an equivalence—hence any object $x\in\mathfrak{S}({\mathscr{S}})$ is isomorphic to ${\mathscr{Y}}(z)$ with $z$ an object of $\widehat{\mathfrak{S}}({\mathscr{S}})$. But if we have an object $t\in\widehat{\mathfrak{L}}({\mathscr{S}})$ with ${\mathscr{Y}}(t)=x\cong{\mathscr{Y}}(z)$, then the isomorphism must lift to $\widehat{\mathfrak{L}}({\mathscr{S}})$, and hence $t\cong z$ must belong to the replete subcategory $\widehat{\mathfrak{S}}({\mathscr{S}})\subset\widehat{\mathfrak{L}}({\mathscr{S}})$.

Now recall the Yoneda embedding $\widetilde{Y}:{\mathscr{S}}\longrightarrow\mathfrak{L}({\mathscr{S}})\subset\text{\rm Mod--}{\mathscr{S}}$. We have the triangle of functors

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which commutes up to natural isomorphism. It immediately follows that

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The first equality is because the inverse images under the isomorphic functors \widetilde{Y}\simeq F\circ\big{[}{\mathscr{Y}}|_{\widehat{\mathfrak{L}}({\mathscr{S}})}\big{]}, of the replete subcategory $\mathfrak{S}({\mathscr{S}})$, must be equal. And the second equality comes from the paragraph above, which informs us that {\mathscr{Y}}^{-1}_{\widehat{\mathfrak{L}}({\mathscr{S}})}\big{[}\mathfrak{S}({\mathscr{S}})\big{]}=\widehat{\mathfrak{S}}({\mathscr{S}}).

Rewriting the second paragraph of the current Remark, by appealing to the equality \widetilde{Y}^{-1}\big{[}\mathfrak{S}({\mathscr{S}})\big{]}=F^{-1}\big{[}\widehat{\mathfrak{S}}({\mathscr{S}})\big{]}, we deduce first that \widetilde{Y}^{-1}\big{[}\mathfrak{S}({\mathscr{S}})\big{]} is a triangulated subcategory of ${\mathscr{S}}$, and then that the functor $\widetilde{Y}:{\mathscr{S}}\longrightarrow\text{\rm Mod--}{\mathscr{S}}$ restricts to a fully faithful, triangulated functor \widetilde{Y}^{-1}\big{[}\mathfrak{S}({\mathscr{S}})\big{]}\longrightarrow\mathfrak{S}({\mathscr{S}}).

All of the discussion above assumed we were in the presence of a good extension $F:{\mathscr{S}}\longrightarrow{\mathscr{T}}$. But the assertion of the last paragraph turns out to be robust. Even though the category $\mathfrak{L}({\mathscr{S}})$ is rarely triangulated it contains both ${\mathscr{S}}$ and $\mathfrak{S}({\mathscr{S}})$ as subcategories—in the case of $\mathfrak{S}({\mathscr{S}})$ this is by definition, while for ${\mathscr{S}}$ we commit the notational crime of confusing ${\mathscr{S}}$ with its its essential image under $\widetilde{Y}:{\mathscr{S}}\longrightarrow\mathfrak{L}({\mathscr{S}})$. Each of ${\mathscr{S}}$ and $\mathfrak{S}({\mathscr{S}})$ has its own triangulated structure. And it is always true that ${\mathscr{S}}\cap\mathfrak{S}({\mathscr{S}})$ has a (unique) triangulated structure so that each of the two embeddings, into ${\mathscr{S}}$ and into $\mathfrak{S}({\mathscr{S}})$, is triangulated. **

The import of Theorem 20 and Remark 21 is that any good extension of ${\mathscr{S}}$ contains both ${\mathscr{S}}$ and $\mathfrak{S}({\mathscr{S}})$ as triangulated subcategories, and the embedding of $\mathfrak{S}({\mathscr{S}})$ into ${\mathscr{T}}$ is explicit enough to facilitate computations, both of $\mathfrak{S}({\mathscr{S}})$ and of ${\mathscr{S}}\cap\mathfrak{S}({\mathscr{S}})$. The author will be the first to admit that better computational tools would be wonderful—this is all we have right now.

Notwithstanding the current limitations on what we know, Theorem 20 does produce interesting examples. We give a few.

Let $H:{\mathscr{S}}\longrightarrow\text{\rm Ab}$ be the homological functor which takes a spectrum $(X,n)$ to its zeroth stable homotopy group; in the notation above this means

[TABLE]

where $\mathbb{S}^{k}$ is the $k$–dimensional sphere. In the standard notation of homotopy theorists $H(X,n)=\pi_{0}(\Sigma^{n}X)$ and $H^{i}(X)=\pi_{-i}(\Sigma^{n}X)$, where $\pi_{-i}$ is the $(-i)^{\hbox{\rm\tiny th}}$ stable homotopy group. Now let the good metric be as in Example 12(i).

Let $F:{\mathscr{S}}\longrightarrow{\mathscr{T}}$ be the embedding of the homotopy category of finite spectra into the homotopy category of all spectra—the homotopy category of all spectra is not quite so easy to describe simply, hence let us leave this out. For us what’s important is that the functor $F$ can be shown to be a good extension. And the computation of $\widehat{\mathfrak{S}}({\mathscr{S}})$, which by Theorem 20 is canonically triangle equivalent to $\mathfrak{S}({\mathscr{S}})$, can be carried out. It shows that $\widehat{\mathfrak{S}}({\mathscr{S}})\subset{\mathscr{T}}$ is given by the formula

[TABLE]

The assertions in the paragraphs above follow from the far more general [Neeb, Example 4.2].

With ${\mathscr{S}}$ still as above, it’s known that ${\mathscr{S}}\cap\widehat{\mathfrak{S}}({\mathscr{S}})\cong{\mathscr{S}}\cap\mathfrak{S}({\mathscr{S}})=\{0\}$ and that ${\mathscr{S}}$ and $\mathfrak{S}({\mathscr{S}})$ are not triangle equivalent. Let us recall.

Let $\mathbb{S}\in{\mathscr{S}}$ be the zero-sphere; in the notation of the first paragraph of the current Example this means $\mathbb{S}=(\mathbb{S}^{n},-n)$ where $n>0$ is an integer and $\mathbb{S}^{n}$ is the $n$–dimensional sphere. And let $K({\mathbb{Z}},0)$ be the Eilenberg-MacLane spectrum defined by

[TABLE]

Next we adopt the terminology of Bondal and Van den Bergh [BVdB03, 2.1]: an object $G$ in a triangulated category ${\mathscr{R}}$ is a classical generator if the smallest thick subcategory containing $G$ is all of ${\mathscr{R}}$. It’s not difficult to see that $\mathbb{S}$ is a classical generator for ${\mathscr{S}}$ while $K({\mathbb{Z}},0)$ is a classical generator for $\widehat{\mathfrak{S}}({\mathscr{S}})$. But it’s also known that

[TABLE]

and that, for all $n\in{\mathbb{Z}}$, the modules above are finitely generated ${\mathbb{Z}}$–modules. The vanishing assertions are ancient, and the finite generation was proved in Serre’s 1951 PhD thesis [Ser51]. Somewhat more recent is the computation that

[TABLE]

while {\mathop{\rm Hom}}\big{(}K({\mathbb{Z}},0),\Sigma\mathbb{S}\big{)} is a ${\mathbb{Q}}$–vector space: the reader can find this in Lin’s 1976 article [Lin76, Theorem 3.6]. The interested reader can look at Margolis 1974 article [Mar74] for an (independent) approach to results similar to Lin’s, and at Ravenel [Rav84, Section 4] for later developments and extensions. Anyway: because $\mathbb{S}$ and $K({\mathbb{Z}},0)$ classically generate the respective subcategories, we immediately deduce

[TABLE]

while for all $n$ we have

[TABLE]

What’s more it’s known that ${\mathop{\rm Hom}}(G,\Sigma^{n}G)\neq 0$ for infinitely many $n$, when $G$ is either $\mathbb{S}$ or $K({\mathbb{Z}},0)$; these estimates on the non-vanishing of Hom-sets can also be found in Serre [Ser51]. It immediately follows that the categories ${\mathscr{S}}$ and $\widehat{\mathfrak{S}}({\mathscr{S}})$ cannot be triangle equivalent, and that ${\mathscr{S}}\cap\widehat{\mathfrak{S}}({\mathscr{S}})\cong{\mathscr{S}}\cap\mathfrak{S}({\mathscr{S}})=\{0\}$. **

Let ${\mathbf{D}}(R\text{--\rm Mod})$ be the unbounded derived category of all complexes of $R$–modules. Then the natural inclusion F:\big{[}{\mathbf{D}}^{b}(\mathop{R\textup{--mod}})\big{]}^{\hbox{\rm\tiny op}}\longrightarrow\big{[}{\mathbf{D}}(R\text{--\rm Mod})\big{]}^{\hbox{\rm\tiny op}} is a good extension with respect to the metric. And this can be used to compute \widehat{\mathfrak{S}}\Big{(}\big{[}{\mathbf{D}}^{b}(\mathop{R\textup{--mod}})\big{]}^{\hbox{\rm\tiny op}}\Big{)} as a subcategory of \big{[}{\mathbf{D}}(R\text{--\rm Mod})\big{]}^{\hbox{\rm\tiny op}}. It turns out to be \big{[}{\mathscr{H}}^{0}\big{(}\text{\bf Perf}(R)\big{)}\big{]}^{\hbox{\rm\tiny op}}, where {\mathscr{H}}^{0}\big{(}\text{\bf Perf}(R)\big{)} is the derived category whose objects are bounded complexes of finitely generated, projective $R$–modules. In this particular case there is an inclusion—and Remark 21 tells us that, with ${\mathscr{S}}={\mathbf{D}}^{b}(\mathop{R\textup{--mod}})$, the subcategories ${\mathscr{S}}^{\hbox{\rm\tiny op}}$ and $\mathfrak{S}({\mathscr{S}}^{\hbox{\rm\tiny op}})$ of $\mathfrak{L}({\mathscr{S}}^{\hbox{\rm\tiny op}})$ satisfy $\mathfrak{S}({\mathscr{S}}^{\hbox{\rm\tiny op}})\subset{\mathscr{S}}^{\hbox{\rm\tiny op}}$. Furthermore this inclusion respects the triangulated structure.

The assertions in the paragraph above follow from the far more general [Neeb, Proposition 5.6].

Thus out of the category ${\mathbf{D}}^{b}(\mathop{R\textup{--mod}})$ we have cooked up its triangulated subcategory {\mathscr{H}}^{0}\big{(}\text{\bf Perf}(R)\big{)}, and hence we also know the quotient

[TABLE]

where ${\mathbf{D}}_{\text{\rm sing}}(R)$ is what’s known in the literature as the singularity category of $R$.

OK: the paragraphs above showed that, given the category ${\mathbf{D}}^{b}(\mathop{R\textup{--mod}})$ and its metric, then out of the data we can construct the triangulated subcategory {\mathscr{H}}^{0}\big{(}\text{\bf Perf}(R)\big{)} and the quotient ${\mathbf{D}}_{\text{\rm sing}}(R)$. The reader might naturally ask if there is a way to construct the metric without appealing to the homological functor $H:{\mathbf{D}}^{b}(\mathop{R\textup{--mod}})\longrightarrow(\mathop{R\textup{--mod}})$. The answer turns out to be Yes up to equivalence. The equivalence class of the metric can be obtained without using anything other than the triangulated structure on ${\mathbf{D}}^{b}(\mathop{R\textup{--mod}})$—we will say a little more about this in Remark 24.

It is also possible to construct examples where the inclusion goes the other way, that is ${\mathscr{S}}\subset\mathfrak{S}({\mathscr{S}})$. In fact: starting with the triangulated category {\mathscr{S}}={\mathscr{H}}^{0}\big{(}\text{\bf Perf}(R)\big{)} and the homological functor H:{\mathscr{H}}^{0}\big{(}\text{\bf Perf}(R)\big{)}\longrightarrow(\mathop{R\textup{--mod}}), the functor taking a cochain complex to its zeroth cohomology module, we can endow ${\mathscr{S}}$ with the good metric of Example 12(i). And then it may be computed that $\mathfrak{S}({\mathscr{S}})={\mathbf{D}}^{b}(\mathop{R\textup{--mod}})$, and as subcategories of $\mathfrak{L}({\mathscr{S}})$ we have an inclusion ${\mathscr{S}}\subset\mathfrak{S}({\mathscr{S}})$ which agrees with the standard triangulated inclusion {\mathscr{H}}^{0}\big{(}\text{\bf Perf}(R)\big{)}\subset{\mathbf{D}}^{b}(\mathop{R\textup{--mod}}). Once again: the equivalence class of the metric has an intrinsic description, it depends only on the triangulated structure of {\mathscr{S}}={\mathscr{H}}^{0}\big{(}\text{\bf Perf}(R)\big{)}. Thus {\mathscr{H}}^{0}\big{(}\text{\bf Perf}(R)\big{)} also contains enough data to determine ${\mathbf{D}}_{\text{\rm sing}}(R)$. See Remark 24 for some elaboration, and [Neeb] for infinitely more detail and much greater generality. **

Let ${\mathscr{S}}$ be a triangulated category and let $G\in{\mathscr{S}}$ be a classical generator—recall: classical generators were defined in Bondal and Van den Bergh [BVdB03, 2.1], and we have already met them in Example 22. The functor ${\mathop{\rm Hom}}(G,-)$ is a homological functor, and we may apply the formulas of Example 12 to obtain metrics. It is an easy exercise to show that, if $G,G^{\prime}\in{\mathscr{S}}$ are two classical generators, then the metrics obtained by applying Example 12 (i), (ii) or (iii) to the functor ${\mathop{\rm Hom}}(G,-)$ are (respectively) equivalent to the metrics obtained by applying Example 12 (i), (ii) or (iii) to the functor ${\mathop{\rm Hom}}(G^{\prime},-)$. Thus on any triangulated category with a classical generator there are three intrinsic equivalence classes of metrics. For the sake of definiteness let us call them ${\mathscr{L}}\!\text{\it ength}_{(i)}^{\mathrm{Int}}$ ${\mathscr{L}}\!\text{\it ength}_{(ii)}^{\mathrm{Int}}$ and ${\mathscr{L}}\!\text{\it ength}_{(iii)}^{\mathrm{Int}}$. To spell it out explicitly: to obtain a metric in the equivalence class ${\mathscr{L}}\!\text{\it ength}_{(i)}^{\mathrm{Int}}$ you choose a classical generator $G$ for the category ${\mathscr{S}}$, and then the formula of Example 12(i), applied to the homological functor ${\mathop{\rm Hom}}(G,-)$, delivers your metric.

The category {\mathscr{H}}^{0}\big{(}\text{\bf Perf}(R)\big{)} is known to have a classical generator, more precisely the object R\in{\mathscr{H}}^{0}\big{(}\text{\bf Perf}(R)\big{)} is a classical generator. To obtain a metric in the equivalence class ${\mathscr{L}}\!\text{\it ength}_{(i)}^{\mathrm{Int}}$ we begin by choosing $R$ for our classical generator, and then we apply the formula of Example 12(i) to the homological functor ${\mathop{\rm Hom}}(R,-)$. But there is a canonical isomorphism $H(-)={\mathop{\rm Hom}}(R,-)$, between of the functor $H$ of final paragraph of Example 23 and the functor ${\mathop{\rm Hom}}(R,-)$. Hence the metric of the final paragraph of Example 23 is in the equivalence class ${\mathscr{L}}\!\text{\it ength}_{(i)}^{\mathrm{Int}}$, it is intrinsic up to equivalence.

We also asserted that the metric on \big{[}{\mathbf{D}}^{b}(\mathop{R\textup{--mod}})\big{]}^{\hbox{\rm\tiny op}}, which was studied in the opening paragraphs of Example 23, is intrinsic up to equivalence. Here the argument is subtler, partly because (as far as I know) the category ${\mathbf{D}}^{b}(\mathop{R\textup{--mod}})$ does not have to have a classical generator. The main idea on how to define the metric intrinsically is presented in the opening paragraphs of the introduction to [Neeb].

There are cases in which the category ${\mathbf{D}}^{b}(\mathop{R\textup{--mod}})$ is known to have a classical generator, and moreover this classical generator is known to satisfy some additional nice properties: it is “strong”—whatever that means—and one has further technical knowledge about it. When this happens to be the case the intrinsic description of the metric simplifies. Concretely: if the ring $R$ is commutative, noetherian, has finite Krull-dimension and every closed subset of ${\text{\rm Spec}(R)}$ admits a regular alteration, then ${\mathbf{D}}^{b}(\mathop{R\textup{--mod}})$ has a classical generator, and the metric on \big{[}{\mathbf{D}}^{b}(\mathop{R\textup{--mod}})\big{]}^{\hbox{\rm\tiny op}} given in the first paragraph of Example 23 is in the equivalence class ${\mathscr{L}}\!\text{\it ength}_{(ii)}^{\mathrm{Int}}$. **

As in the article [Neeb], Krause [Kra] comes up with a procedure that produces out of ${\mathscr{S}}$ the category we call $\mathfrak{L}({\mathscr{S}})$—Krause’s recipe is different from Definition 5(i), but out of a different oven comes exactly the same dish. For the reader interested in looking up Krause [Kra] for more detail: the category we call $\mathfrak{L}({\mathscr{S}})$ is canonically equivalent to what goes by the name $\widehat{\mathscr{S}}$ in [Kra]. And just as in [Neeb] Krause looks at the special case where {\mathscr{S}}={\mathscr{H}}^{0}\big{(}\text{\bf Perf}(R)\big{)} as in Example 23. With H:{\mathscr{H}}^{0}\big{(}\text{\bf Perf}(R)\big{)}\longrightarrow(\mathop{R\textup{--mod}}) the homological functor of the last paragraphs of Example 23, Krause studies the metric of Example 12(iii)—this is where his treatment radically differs from Example 23, where our metric was the one of Example 12(i). Because we now have two metrics let us denote the completion of Example 23 by $\mathfrak{L}_{1}({\mathscr{S}})$ and Krause’s completion by $\mathfrak{L}_{2}({\mathscr{S}})$. It isn’t difficult to show that, inside the category $\text{\rm Mod--}{\mathscr{S}}$, there is an inclusion $\mathfrak{L}_{1}({\mathscr{S}})\subset\mathfrak{L}_{2}({\mathscr{S}})$.

Of course the category $\mathfrak{S}_{1}({\mathscr{S}})\cong{\mathbf{D}}^{b}(\mathop{R\textup{--mod}})$, being a full subcategory of $\mathfrak{L}_{1}({\mathscr{S}})$, is also a full subcategory of $\mathfrak{L}_{2}({\mathscr{S}})$. And there is an intrinsic description of the subcategory $\mathfrak{S}_{1}({\mathscr{S}})\subset\mathfrak{L}_{2}({\mathscr{S}})$, the reader can find it in [Kra]. Krause denotes it $\widehat{\mathscr{S}}^{b}$, I suppose in our notation it should be $\mathfrak{L}_{2}({\mathscr{S}})^{b}$.

Next we come to the triangles. With respect to the metric of Example 12(i) the triangles in $\mathfrak{S}_{1}({\mathscr{S}})$ are simply the colimits of Cauchy sequences of triangles in ${\mathscr{S}}$—subject of course to the restriction that the colimit lies in $\mathfrak{S}_{1}({\mathscr{S}})\subset\mathfrak{L}_{1}({\mathscr{S}})$. Theorem 15 tells us that, with the triangles defined as above, the category $\mathfrak{S}_{1}({\mathscr{S}})$ is triangulated.

Now let’s work out some consequences. Let $F^{\prime}\longrightarrow F$ be any morphism in $\mathfrak{S}_{1}({\mathscr{S}})$; it can be expressed as the colimit of a Cauchy sequence of morphisms $s^{\prime}_{*}\longrightarrow s_{*}$ in ${\mathscr{S}}$, where the metric on ${\mathscr{S}}$ is as in Example 12(i). We may (non-canonically) complete this to a sequence of triangles $s^{\prime}_{*}\longrightarrow s_{*}\longrightarrow s^{\prime\prime}_{*}\longrightarrow\Sigma s^{\prime}_{*}$ in ${\mathscr{S}}$, and the reader can easily check that this sequence is Cauchy in the metric of Example 12(i) and the colimit lies in $\mathfrak{S}_{1}({\mathscr{S}})$. By Definition 13 the colimit is a distinguished triangle in $\mathfrak{S}_{1}({\mathscr{S}})$ extending the morphism $F^{\prime}\longrightarrow F$. But from the axioms of triangulated categories the extension of the morphism $F^{\prime}\longrightarrow F$ to a distinguished triangle in $\mathfrak{S}_{1}({\mathscr{S}})$ is unique up to non-canonical isomorphism. It follows that the sequences of triangles in ${\mathscr{S}}$ of the form $s^{\prime}_{*}\longrightarrow s_{*}\longrightarrow s^{\prime\prime}_{*}\longrightarrow\Sigma s^{\prime}_{*}$, extending the given Cauchy sequence of morphisms $s^{\prime}_{*}\longrightarrow s_{*}$, must all be non-canonically Ind-isomorphic.

This can be proved, but the proof I know relies heavily on the fact that the colimit lies in the subcategory $\mathfrak{S}_{1}({\mathscr{S}})\subset\mathfrak{L}_{1}({\mathscr{S}})$, which was chosen carefully in terms of the metric. Whereas Krause’s definition of $\mathfrak{L}_{2}({\mathscr{S}})^{b}\subset\mathfrak{L}_{2}({\mathscr{S}})$ makes no mention of the metric.

Now let’s compare the Cauchy sequences with respect to the two metrics under consideration. Since the metric of Example 12(i) is finer than the metric of Example 12(iii) there are more Cauchy sequences with respect to Krause’s metric—this is what leads to the (proper) inclusion $\mathfrak{L}_{1}({\mathscr{S}})\subset\mathfrak{L}_{2}({\mathscr{S}})$. As it turns out any Cauchy sequence of morphisms $s^{\prime}_{*}\longrightarrow s_{*}$, with respect to the metric of Example 12(iii) and whose colimit happens to lie in $\mathfrak{L}_{1}({\mathscr{S}})\subset\mathfrak{L}_{2}({\mathscr{S}})$, is Ind-isomorphic to a Cauchy sequence $t^{\prime}_{*}\longrightarrow t_{*}$ with respect to the metric of Example 12(i). So we might be tempted to guess that the Cauchy sequences of triangles with respect to the metric of Example 12(iii), with colimits in $\mathfrak{S}_{1}({\mathscr{S}})=\mathfrak{L}_{2}({\mathscr{S}})^{b}$, will also be Ind-isomorphic to Cauchy sequences of triangles with respect to the metric of Example 12(i). But if we try to produce such an Ind-isomorphism we run into the problem that the mapping cone isn’t functorial—the simple-minded approach breaks down. As the definition of $\mathfrak{L}_{2}({\mathscr{S}})^{b}\subset\mathfrak{L}_{2}({\mathscr{S}})$ doesn’t involve the metric I see no sophisticated alternative to the simple-minded method—for all I know there might be Cauchy sequences of triangles, with respect to the metric of Example 12(iii) and with colimit in $\mathfrak{S}_{1}({\mathscr{S}})$, which aren’t Ind-isomorphic to Cauchy sequences of triangles with respect to the metric of Example 12(i).

Krause’s solution to the problem is to fix an enhancement, and only admit those Cauchy sequences that lift to the chosen enhancement. For the situation at hand a minimal enhancement suffices—it’s enough to assume we are working with Keller’s towers, see Keller [Kel91] for the original exposition, or his appendix to Krause [Kra] for a condensed version.

Of course it is possible to apply the machinery surveyed here to Krause’s metric—we obtain a triangulated category $\mathfrak{S}_{2}({\mathscr{S}})$ whose triangulated structure is enhancement-free. Using a good extension with respect to the metric and Theorem 20 it can be computed that $\mathfrak{S}_{2}({\mathscr{S}})$ is a proper subcategory of $\mathfrak{S}_{1}({\mathscr{S}})\cong{\mathbf{D}}^{b}(\mathop{R\textup{--mod}})$—the objects are those complexes in ${\mathbf{D}}^{b}(\mathop{R\textup{--mod}})$ which have bounded injective resolutions. For more detail the reader is referred to [Neeb, Example 4.9]. **

We have said it before but repeat for emphasis: Theorem 20 is at present the only computational tool we have. It would be great to have some more ways to compute $\mathfrak{S}({\mathscr{S}})$.

Acknowledgements.   The author would like to thank Henning Krause for asking the question that led to the research we have surveyed in this note, which is expounded more fully and in greater generality in [Neeb]. The author also wants to thank Steve Lack for pointing him to Lawvere’s old paper [Law73, Law02], Richard Garner for explaining the relevance of Betti and Galuzzi [BG75], Kelly [Kel82], Kelly and Schmitt [KS05] and Kubiś [Kub], as well as Paul Balmer, Ivo Dell’Ambrogio, David Gepner, Bernhard Keller, Sasha Kuznetsov, Tom Leinster, Nori Minami, Doug Ravenel and Greg Stevenson for corrections and helpful comments on earlier drafts of this manuscript and help with references to other relevant literature.