The n-Category Café

But for the short version: a prederivator is a strict 2-functor $\mathbb{D}\colon\mathbf{Cat}^\text{op}\to\mathbf{CAT}$. For $J,K\in\mathbf{Cat}$ and a functor $u\colon J\to K$, we obtain a restriction functor $u^\ast\colon\mathbb{D}(K)\to\mathbb{D}(J)$. A natural transformation $\alpha\colon u\to v$ is mapped contravariantly to $\alpha^\ast\colon u^\ast\to v^\ast$.

We think of the domain of a derivator as diagrams in the shape of small categories, and the outputs coherent diagrams with values in some category $\mathcal{C}$. Specifically, that category $\mathcal{C}$ is the value of $\mathbb{D}$ at the one-point category $e$. We’ll call $\mathbb{D}(e)$ the underlying category of $\mathbb{D}$.

A derivator is a prederivator satisfying some more axioms, most pertinent among them being the following: each $u^\ast$ admits a left adjoint $u_!$ and a right adjoint $u_\ast$ (called the homotopy Kan extensions along $u$). As a consequence, $\mathbb{D}(e)$ admits all homotopy limits and homotopy colimits. For one final axiom, a pointed derivator is a derivator with a pointed base, which implies that each $\mathbb{D}(K)$ is also pointed. In this situation, we obtain a suspension-loop adjunction on $\mathbb{D}(e)$, where suspension $\Sigma$ is defined as the (homotopy) pushout of $0\leftarrow X\rightarrow 0$ and loop $\Omega$ is the (homotopy) pullback of $0\rightarrow X\leftarrow 0$. It’s worth emphasizing that, once we have a derivator whose base happens to be pointed, the suspension and loop are canonical adjoints and are determined by the higher structure of the derivator. Other interesting properties are also automatically satisfied, but we needn’t get into that here.

What examples in real life give rise to a pointed derivator? For any pointed combinatorial model category $\mathcal{M}$, we can consider its homotopy category $\operatorname{Ho}\mathcal{M}$. It can be shown that, for any small category $K$, $\mathcal{M}^K$ admits both a projective and an injective model structure. This leads us to a pointed derivator $\mathbb{D}_{\mathcal{M}}$, defined by $\mathbb{D}_{\mathcal{M}}(K):=\operatorname{Ho}(\mathcal{M}^K)$. For $\mathcal{C}$ a pointed $\infty$-category, we have a similar derivator $\mathbb{D}_{\mathcal{C}}(K):=\operatorname{Ho}(\mathcal{C}^{N(K)})$, where $N(K)$ is the nerve of $K$.

The next question is of stability. A pointed derivator is stable if the $(\Sigma,\Omega)$ adjunction is an adjoint equivalence. The stable homotopy category of pointed spaces is constructed for precisely this purpose, that is, so that reduced suspension and loop space become inverse equivalences. There are many (classical) ways to perform this stabilization, and we aim for the abstract formulation in derivators.

If we are given a pointed derivator $\mathbb{D}$, is there some universal nearest stable derivator? Put another way, is there some derivator $\operatorname{St}\mathbb{D}$ so that morphisms from $\mathbb{D}$ into a stable derivator $\mathbb{S}$ are the “same” as morphisms $\operatorname{St}\mathbb{D}\to \mathbb{S}$? The answer is yes, and we have the following strong result.

Theorem 7.14. Let $\mathbf{Der}_!$ be the 2-category with objects regular pointed derivators, maps cocontinuous morphisms of derivators, and natural transformations modifications. Let $\mathbf{StDer}_!$ be the full sub-2-category of stable derivators. There is a pseudofunctor $\operatorname{St}\colon\mathbf{Der}_!\to\mathbf{StDer}_!$ which is left adjoint to the inclusion. Specifically, there is a universal cocontinuous morphism of derivators $$\operatorname{stab}\colon\mathbb{D}\to\operatorname{St}\mathbb{D}$$ and precomposition with this morphism give an equivalence of categories of cocontinuous morphisms $$\operatorname{Hom}_!(\operatorname{St}\mathbb{D},\mathbb{S})\overset{\operatorname{stab}^\ast}{\longrightarrow}\operatorname{Hom}_!(\mathbb{D},\mathbb{S})$$ for any stable derivator $\mathbb{S}$.

The adjective regular will be explained in a few paragraphs. To give a sketch of the proof, we first develop a theory of prespectrum objects in a derivator $\mathbb{D}$ (Notation 5.8). Consider the poset $V\subset\mathbb{Z}^2$ on the objects $(i,j)$ such that $|i-j|\leq 1$. We construct a pointed derivator which we name $\operatorname{Sp}\mathbb{D}$ whose underlying category is the subcategory of $\mathbb{D}(V)$ that vanish off the diagonal. That this is a pointed derivator is a consequence of Theorem 4.10, which was stated without proof by Heller (his Proposition 7.4). The proof is nontrivial but straightforward, and it’s a fantastic exercise in the calculus of mates.

An object $X\in\operatorname{Sp}\mathbb{D}(e)$ looks like $$\array{&&0&\to&X_1\\ &&\uparrow&&\uparrow\\ 0&\to&X_0&\to&0\\ \uparrow&&\uparrow&&\\ X_{-1}&\to&0&&}$$ extending infinitely in both directions. The higher structure of the derivator encodes the comparison maps $\sigma_n\colon X_n\to \Omega X_{n+1}$ for all $n\in\mathbb{Z}$ naturally in such an object, so we rightfully call such an $X$ a prespectrum object. If all these comparison maps are isomorphisms, we call $X$ a (stable) spectrum object. We let $\operatorname{St}\mathbb{D}\subset\operatorname{Sp}\mathbb{D}$ be the full subprederivator (i.e. full subcategories at each $K\in\mathbf{Cat}$) consisting of the stable spectrum objects.

There is no guarantee that the stable spectrum prederivator is actually a derivator, let alone a stable derivator. The first big theorem of the paper (Theorem 6.12) is to show that there is a localization $\operatorname{Sp}\mathbb{D}\to\operatorname{St}\mathbb{D}$. Lemme 4.2 of Denis-Charles Cisinski in Catégories Dérivables says that the localization of any derivator is still a derivator, so that takes care of the first concern.

To prove that $\operatorname{St}\mathbb{D}$ is stable, we now need to define the adjective regular. A derivator is called regular if filtered colimits commute with finite limits. The derivator associated to any $n$-topos for $n\in[0,\infty]$ is regular, as is the derivator associated to a Grothendieck abelian category. One of the equivalent definitions of stability for a derivator is that all colimits commute with finite limits, so regularity is a kind of pre-stability assumption on $\mathbb{D}$. In this case, $\operatorname{St}\mathbb{D}$ is stable (Lemma 6.19 and Proposition 6.23). (As a side note: I am in the market for an alternative to regular as a descriptor for this situation and welcome audience suggestions.)

The last thing to settle is the universal property of the stabilization. Most specifically, the pseudonaturality of $\operatorname{St}$ on morphisms in the domain. Heller’s original proof has a critical error that we take great pains to fix in §7 of the paper, specifically in Lemma 7.4. Heller writes down a diagram incoherently (his Diagram 9.3) that he lifts to a coherent object. Perhaps the crowning achievement of the paper is the construction of this diagram coherently, which takes place through a subposet of $\mathbb{Z}^5$. Diagram yoga is absent in Heller’s original work, but our proof methodology has a diagrammatic flavor that is emblematic of the theory of derivators in modern scholarship. This makes it difficult to recreate the more ambitious diagrams within this blog, and so interested readers should check out the paper itself. Specifically, most of the subsections titled “Construction” repair baseless (though ultimately provable) claims by Heller and contain the most complex diagram shapes.