# What are _all_ of the exactness properties enjoyed by stable $\infty$-categories?

Alternate formulation of the question (I think): What's a precise version of the statement: "In a stable $\infty$-category, finite limits and finite colimits coincide"?

Recall that a stable $\infty$-category is a type of finitely complete and cocomplete $\infty$-category characterized by certain exactness conditions. Namely,

1. There is a zero object $0$, i.e. an object which is both initial and terminal.

2. Every pushout square is a pullback square and vice versa.

Item (2) takes advantage of a peculiar symmetry of the "square" category $S = \downarrow^\to_\to \downarrow$; namely $S$ can either be regarded as $S' \ast \mathrm{pt}$ where $S' = \cdot \leftarrow \cdot \rightarrow \cdot$ is the universal pushout diagram, or $S$ can be regarded as $S = \mathrm{pt} \ast S''$ where $S'' = \cdot \rightarrow \cdot \leftarrow \cdot$ is the universal pullback diagram. Hence it makes sense to ask, for a given $S$-diagram, whether it is a pullback, a pushout, or both. Item (1) similarly takes advantages of the identities $\mathrm{pt} = \emptyset \ast \mathrm{pt} = \mathrm{pt} \ast \emptyset$.

But I can't shake the feeling that notion of a stable infinity category somehow "transcends" this funny fact about the geometry of points and squares. For one thing, one can use a different "combinatorial basis" to characterize the exactness properties of a stable $\infty$-category, namely:

1.' The category is (pre)additive (i.e. finite products and coproducts coincide)

2.' The loops / suspension adjunction is an equivalence.

True, (2') may be regarded as a special case of (2) -- but it may also be regarded as a statement about the (co)tensoring of the category in finite spaces.

Both of these ways of defining stability say that certain limits and colimits "coincide", and my sense is that in a stable $\infty$-category, all finite limits and colimits coincide -- insofar as this makes sense.

Question:

Is there a general notion of "a limit and colimit coinciding" which includes

• zero objects

• biproducts (= products which are also coproducts)

• squares which are both pullbacks and pushouts

• suspensions which are also deloopings

and if so, is it true that in in a stable $\infty$-category, finite limits and finite colimits coincide whenever this makes sense?

I would regard this as investigating a different sort of exactness to the exactness properties enjoyed by ($\infty$)-toposes. In the topos case, I think there are some good answers. For one, in a topos $C$, the functor $C^\mathrm{op} \to \mathsf{Cat}$, $X \mapsto C/X$ preserves limits. Foir another, a Grothendieck topos $C$ is what Street calls "lex total": there is a left exact left adjoint to the Yoneda embedding. It would be nice to have similar statements here which in some sense formulate a "maximal" list of exactness properties enjoyed by (presentable, perhaps) stable $\infty$-categories, rather than the "minimal" lists found in (1,2) and (1',2') above.

I don't think this "finite limits and finite colimits coincide" business can be taken very far. If you take any small category $S_0$ you can add an initial and a terminal object to form $S = \mathrm{pt} \ast S_0 \ast \mathrm{pt}$. A diagram of shape $S$ could potentially be both a colimiting cocone and a limiting cone and you might hope those conditions are equivalent in a stable $\infty$-category. (And for $S_0 = \mathrm{pt} \sqcup \mathrm{pt}$, this does happen, of course: it is the condition that a square is a pushout if and only if it is a pullback.) But this fails1 for three points, $S_0 = \mathrm{pt} \sqcup \mathrm{pt} \sqcup \mathrm{pt}$.

I think it's probably better to focus on a lesser known characterization of stable $\infty$-categories: they are precisely the finitely complete and cocomplete ones in which finite limits commute with finite colimits.2

1 The colimit of the $\mathrm{pt} \ast S_0$ shaped diagram with $X$ at the cone point and zeroes in the other slots is $\Sigma X \amalg \Sigma X$. Analogously the limit of the $S_0 \ast \mathrm{pt}$ diagram with $Y$ in the cocone point and zeroes in the other slots is $\Omega Y \times \Omega Y$. But for $Y = \Sigma X \amalg \Sigma X$ we do not have $X = \Omega Y \times \Omega Y$.

2 If $\mathcal{C}$ is a stable $\infty$-category, then it is finitely cocomplete, and thus if $S$ is a finite diagram shape, there is a functor $\mathrm{colim} : \mathrm{Fun}(S, \mathcal{C}) \to \mathcal{C}$. It's domain is also stable and $\mathrm{colim}$ preserves finite colimits --because colimits commute with colimits. The functor is therefore exact and so preserves finite limits as well.

Now, assume that $\mathcal{C}$ is finitely complete and cocomplete and that finite limits commute with colimits in it. Consider the following diagram: $$\require{AMScd}\begin{CD} X @<<< X @>>> 0 \\ @VVV @VVV @VVV \\ 0 @<<< X @>>> 0 \\ @AAA @AAA @AAA \\ 0 @<<< X @>>> X \\ \end{CD}$$ Taking pushouts of the rows we get the diagram $0 \to \Sigma X \leftarrow 0$ whose pullback is $\Omega \Sigma X$. If instead we take pullbacks of the columns, we get $X \leftarrow X \to X$, whose pushout is $X$. Pullbacks commuting with pushouts tell us then that $\Omega \Sigma X \cong X$ so $\mathcal{C}$ is stable.

• The lesser known characterization is due I believe to Moritz Groth. Apr 15, 2017 at 16:21
• Oh, I didn't know that, @CharlesRezk, thanks! You are probably referring to this paper: Characterizations of abstract stable homotopy theories, right? (I hadn't seen it because, well, I'm a little negligent when it comes to reading papers that use derivators.) I thought of this a "folklore" result, so it's good to have something to cite, instead. Apr 15, 2017 at 17:32
• A slight generalization: if $\phi: S \to T$ is a functor between finite $\infty$-categories and $\mathcal{C}$ is stable, then the left Kan extension $\phi^\ast: \mathcal{C}^S \to \mathcal{C}^T$ is exact because it has a right adjoint given by the direct image $\phi_\ast : \mathcal{C}^T \to \mathcal{C}^S$ (surely this is in Groth's paper, it sounds pretty derivator-y). I wonder if there is a "weighted limit/colimit" version of this statement... Apr 15, 2017 at 18:20
• Oh cool! Groth also notices another thing I'm secretly trying to understand! Namely, if $C$ is stable, then the constant functor $C \to C^{}$ fits into an infinite string of adjoints! This generalizes to all the simplicial maps between $C^{[m]}$ and $C^{[n]}$, I think, but I'm not sure how much further it goes. Apr 15, 2017 at 19:14
• @TimCampion your comment on infinite strings of adjoints made me think to this paper: arxiv.org/pdf/1501.01999.pdf by Balmer, dell'Ambrogio and Sanders, maybe it is of your interest. Oct 23, 2020 at 12:45

Regarding coincidence of limits and colimits, I think the notion you are looking for is that of an absolute (co)limit. When $J$ is absolute for a given enriching ($\infty$-)category $V$ (which in the case of stable $\infty$-categories is the category of spectra), there is another weight $J^*$ such that $J$-weighted colimits naturally coincide with $J^*$-weighted limits.

The sneaky thing is that in some cases it happens that $J^*=J$, so that colimits of a given shape coincide with limits of the same shape. For instance, when $V$ is pointed sets (or spaces) then terminal objects (limits of the empty diagram) coincide with initial objects (colimits of the empty diagram); when $V$ is abelian monoids (or $E_\infty$-spaces) then finite products (limits of finite discrete diagrams) coincide with finite colimits (colimits of finite discrete diagrams); and when $V$ is spectra, suspensions (copowers by $S^1$) coincide with loops (powers by $S^1$). But as Omar points out, this doesn't go as far as you want. Already the "pushout-pullback" coincidence in the stable case is not of this form: the pushout of a span (an ordinary conical colimit) is not the ordinary conical limit of the same span. What is true is that there is a different, non-conical, weight $J^*$ such that the pushout of a span is the $J^*$-weighted limit of that same span.

Charles mentioned Moritz Groth's paper about commutation of finite limits and colimits in stable derivators. Moritz and I are currently working together on a "weighted" generalization of this, whose goal is to reverse the role that the enrichment plays in absoluteness. In the classical theory of absolute (co)limits, the enriching category $V$ is fixed at the outset before we ask which weights are absolute. But in particular examples we can go in the other direction too: from a limit-colimit commutation/coincidence we can construct an enrichment over some "universal" $V$. A category with a zero objects is automatically enriched over pointed sets (or spaces), a category with biproducts is automatically enriched over abelian monoids (or $E_\infty$-spaces), and an $\infty$-category in which finite limits and colimits commute is automatically enriched over spectra. Our first paper (which incorporates most of Moritz's preprint), which is due out any day now, pushes derivators as far as they can go in this direction, which is pretty far but doesn't quite extend to constructing the universal $V$ in general; our plan is to do that with local presentability in a sequel.

• That's really interesting! I had been trying to think about this in terms of absolute colimits (I think inspired by something you once said about enrichment in spectra having all finite limits absolute), but it seemed like it didn't get me anywhere on account of $J^\ast$ being a spectrally enriched weight but not a space-enriched weight. But actually constructing an enrichment sounds like a great idea -- I'll look forward to reading about it! Apr 15, 2017 at 22:38
• This is a beautiful idea, and I very much look forward to reading the paper when it is ready. Is there any relation to the fact that (unenriched) limits can be constructed using large colimits (e.g. Proposition 12.8 of The Joy of Cats)? Apr 26, 2022 at 22:45
• @varkor The first paper is arxiv.org/abs/1704.08084. I don't know if or when the later one will happen -- things got kind of stalled and we got busy with other things. I don't know of a connection to the fact you mention. Apr 27, 2022 at 16:33

The other answers address the question posed in the body of your post. Let me instead answer the question posed in its title: what are all the exactness properties enjoyed by stable $$\infty$$-categories? This answer is inspired by the treatment of abelian categories in Freyd & Scedrov's book Categories, Allegories.

Answer. The exactness properties enjoyed by all stable $$\infty$$-categories are precisely those enjoyed by the $$\infty$$-category of spectra.

Let's try to turn this into a mathematical statement.

Terminology. An $$\infty$$-category is bicartesian if it has all finite limits and finite colimits. A functor between bicartesian $$\infty$$-categories is exact if it preserves finite limits and finite colimits, and is conservative if it reflects isomorphisms. An exactness property is a Horn sentence in the language of bicartesian $$\infty$$-categories. (I don't know if this last definition can be made rigorous, but I hope you understand what meaning I am trying to convey.)

Proposition 1. An $$\infty$$-category is stable iff it is bicartesian and satisfies all exactness properties which hold for the $$\infty$$-category of spectra.

The sufficiency of this condition follows from the observation that the definition of stable $$\infty$$-categories given in your post consists of Horn sentences in the bicartesian predicates ($$0 \to 1$$ is an isomorphism, a pullback square is a pushout square, and vice versa) and the fact that the $$\infty$$-category of spectra is stable.

Since any exactness property involves only a small amount of data, and since exactness properties are reflected by conservative exact functors, the converse follows from the following

Proposition 2. A small $$\infty$$-category is stable iff it is bicartesian and admits a conservative exact functor to the $$\infty$$-category of spectra.

Sufficiency follows as above, since conservative functors reflect the (co)limits they preserve. Necessity follows from the argument: if $$\mathcal{A}$$ is a small stable $$\infty$$-category, then the composite functor $$\mathcal{A} \longrightarrow \mathrm{Fun}(\mathcal{A}^\mathrm{op},\mathbf{Sp}) \longrightarrow \mathbf{Sp}^{\mathrm{ob}\mathcal{A}} \longrightarrow \mathbf{Sp}$$ is conservative and exact since each factor is. The factors are (i) the spectral Yoneda embedding, (ii) evaluation at the objects of $$\mathcal{A}$$, and (iii) the coproduct functor.

The fundamental stability axiom (pushouts are pullbacks and vice-versa) can be seen geometrically as a particular instance of Verider--Lurie duality for the circle. So, Lurie's formulation of Verdier duality (summarized below) for general spaces can be thought of as an answer to your question.

Let $$X$$ be a locally compact Hausdorff space. Given a presheaf $$F$$ on $$X$$ valued in a pointed $$\infty$$-category $$C$$ which sends $$\varnothing\subseteq X$$ to a zero object, we can define a "dual cosheaf" $$\mathbb DF(U)=\operatorname{cofib}(F(X)\to F(X\setminus U))$$ which is a precosheaf on $$X$$ valued in $$C$$ which sends $$\varnothing$$ to a zero object. I will ignore the detail that $$X\setminus U$$ isn't usually an open subset of $$X$$ (but see Lurie HTT 7.3.4 for why this isn't too bad). There is a natural map $$F\to\mathbb D\mathbb DF$$.

When $$C$$ is stable, $$\mathbb D$$ sends sheaves to cosheaves and the natural map $$F\to\mathbb D\mathbb DF$$ is an isomorphism (Lurie HA 5.5.5). It's likely (just by considering very simple compact Hausdorff spaces and very simple (co)sheaves) that the converse is true (if $$\mathbb D$$ sends sheaves to cosheaves then $$C$$ is stable).

Remark: It would indeed be great to replace compact Hausdorff spaces in the above discussion with finite topological spaces. Is there a duality result for finite topological spaces, or perhaps finite posets with their poset topology?

Remark: There should be a notion of "global stable $$\infty$$-category" and the initial such category (with marked object(s)) should be the global stable homotopy category (compatible families of $$G$$-spectra for all compact Lie groups $$G$$). Then I expect there is a form of the above discussion where orbispaces or global spaces or local quotient stacks replace compact Hausdorff spaces.

• Thanks, John, this is super cool! An observation I love which might be related is that if $C$ is stable, then the diagonal functors $C \to C^2$ and $C \to C^{}$ each fit into infinite strings of adjoints (the former adjoint string is periodic of period 2; the latter is periodic up to a shift by the suspension functor, with period 6: $cofib \dashv (0 \to (-)) \dashv cod \dashv id_{(-)} \dashv dom \dashv (-) \to 0 \dashv fib = \Omega cof$. This is somehow very reminiscent of the shifted-periodic adjoint strings appearing in Verdier duality. And this property also characterizes stability... Apr 18, 2022 at 17:46
• So maybe it will suffice to consider sheaves on the Sierpinski space and the 2-point discrete space? Ignoring the fact that the former is not Hausdorff... Apr 18, 2022 at 17:46
• I also can't resist mentioning that there's a shifted-periodic string of period 24 lurking here too -- I don't remember if it's between $C$ and $C^{}$ or $C^{}$ and $C^{}$. (EDIT -- Oh I see that I also mentioned this stuff above.) Apr 18, 2022 at 17:49

Here's an attempt at a weighted version. Let $R,S$ be finite ring-spectra, and let $Y$ be a finite left $R$-module and $Z$ a finite right $S$-module. Let $F$ be a right $R \wedge S$-algebra in a stable $\infty$-category $C$. (So $R$ and $S$ can be regarded as $\infty$-categories, and $F$ is a functor $R \times S \to C$; $Y$ is a presheaf on $R$ and $Z$ is a copresheaf on $X$, both valued in spectra.)

Then we can form a $\Delta \times \Delta^\mathrm{op}$-object in $C$ that looks like this:

$\begin{matrix} (F \wedge Y)^Z & {}^\to_\to & (F\wedge Y)^{Z \wedge S} & \dots \\ \uparrow\uparrow & & \uparrow \uparrow \\ (F \wedge R \wedge Y)^Z & {}^\to_\to & (F\wedge R \wedge Y)^{Z \wedge S} &\dots \\ \vdots & & \vdots & \ddots \end{matrix}$

Now we can take the totalization of the rows and then the realization of the columns, or vice versa. (This is a derived coend computing a weighted homotopy colimit crossed with a derived end computing a weighted homotopy limit; or its a cross of a bar complex and a cobar complex).

At finite stages of the limit / colimit these two will coincide because finite limits commute with finite colimits in $C$. But I'm not sure whether the entire process commutes.