nLab
stable derivator

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Homological algebra

homological algebra

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nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

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diagram chasing

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Stable homotopy theory

Contents

Idea

Just as a derivator is a notion which lies in between a model category or a (∞,1)-category and its homotopy category, a stable (or triangulated) derivator is a notion which lies in between a stable model category or a stable (∞,1)-category and its homotopy triangulated category.

Stable derivators are a useful refinement of triangulated categories, since they contain enough information so that homotopy limit and colimit constructions can be performed “coherently” and desired maps and objects can be specified by true universal properties. This resolves many common issues with triangulated categories stemming from the fact that at the level of a homotopy category, certain desirable maps can only be stipulated to exist with some weak properties, but not characterized precisely among the class of maps that have those properties.

Definition

As is the case for stable (∞,1)-categories, being stable is a property of a derivator rather than a structure (or more precisely, it is a property-like structure). In fact, an (,1)(\infty,1)-category is stable precisely when its underlying derivator is.

Write \square for the “free-living commutative square”

\array{ & \overset{}{\to} & \\ \downarrow && \downarrow\\ & \underset{}{\to} & }

and let LL and RR denote the upper-left span and the lower-right cospan as subcategories of \square, with inclusions i:Li\colon L\to \square and j:Rj \colon R\to \square. For a derivator DD, an object XD()X\in D(\square) is said to be cartesian if the unit Xj *j *XX\to j_* j^* X is an isomorphism, and dually cocartesian if the counit i !i *XXi_! i^* X\to X is an isomorphism. Since ii and jj are fully faithful, it follows from a general theorem about derivators that i !i_! and j *j_* are also fully faithful; thus being cartesian is equivalent to being of the form j *Yj_* Y for some YD(R)Y\in D(R), and likewise being cocartesian is equivalent to being of the form i !Zi_! Z.

A derivator DD is stable (or triangulated) if it is pointed, and moreover an XD()X\in D(\square) is cartesian if and only if it is cocartesian. Such a square is then called bicartesian.

Properties

Triangulation

One of the central facts about stable derivators is:

Theorem (Maltsiniotis)

If DD is a stable derivator, then each category D(X)D(X) is a triangulated category in a canonical way.

We describe the constructions when X=1X=1. The shift/suspension functor S:D(1)D(1)S\colon D(1)\to D(1) is defined by

S=b *i !a * S = b^* i_! a_*

where a:1La\colon 1\to L is the inclusion of the top-left vertex of the span, and b:1b\colon 1\to \square is the inclusion of the bottom-right vertex of the square. In other words, for an object XD(1)X\in D(1) its suspension is defined by the homotopy pushout

X 0 0 SX.\array{X & \overset{}{\to} & 0\\ \downarrow && \downarrow\\ 0 & \underset{}{\to} & S X.}

This makes sense in the more general context of any pointed derivator, but the stability axiom guarantees that SS is actually an equivalence of categories. Its inverse being given by the obvious dual “loop space” construction (which in a general pointed derivator is only right adjoint to it). This provides a motivation for the stability axiom: it is a generalization of the statement that every object is the loop space of its suspension and the suspension of its loop space.

One can also prove that SS is also a copower with the pointed circle S 1S^1 in a suitable sense. In particular, since every object is isomorphic to a double suspension, it is a cogroup object; thus D(X)D(X) is canonically an Ab-enriched category.

Let QQ denote the category

\array{ & \overset{}{\to} & & \to &\\ \downarrow && \downarrow && \downarrow\\ & \underset{}{\to} & & \to &}

with inclusions m,n,p:Qm,n,p\colon \square \to Q of the left and right squares and the outer rectangle, respectively. An object XD(Q)X\in D(Q) is bicartesian if m *Xm^*X, n *Xn^*X, and p *Xp^*X are all bicartesian.

Now consider a bicartesian object XD(Q)X\in D(Q) of the form:

A f B 0 g 0 C h D.\array{A & \overset{f}{\to} & B & \to & 0\\ \downarrow && \downarrow^g && \downarrow\\ 0 & \to & C & \underset{h}{\to} & D.}

In other words, we stipulate that the restrictions to D(1)D(1) along the inclusions of the lower-left and upper-right vertices be zero objects. Now since XX is bicartesian, the outer square

A 0 0 D\array{A & \overset{}{\to} & 0\\ \downarrow && \downarrow\\ 0& \underset{}{\to} & D}

is bicartesian, and thus induces an isomorphism DSAD \cong S A. Thus, from XX we can extract a “triangle”

AfBgChSA A \overset{f}{\to} B \overset{g}{\to} C \overset{h}{\to} S A

and we define the distinguished triangles in D(1)D(1) to be those isomorphic to triangles obtained in this way.

One can then prove the axioms of a triangulated category.

Remark

If AfBgChSA A \overset{f}{\to} B \overset{g}{\to} C \overset{h}{\to} S A is a distinguished triangle in a triangulated category, then AfBgChSA A \overset{f}{\to} B \overset{g}{\to} C \overset{-h}{\to} S A is also an “exact” triangle in the sense that it induces long exact sequences in homology and cohomology, but it is not in general distinguished. On the other hand, if

A f B 0 g 0 C h D.\array{A & \overset{f}{\to} & B & \to & 0\\ \downarrow && \downarrow^g && \downarrow\\ 0 & \to & C & \underset{h}{\to} & D.}

is bicartesian in an (,1)(\infty,1)-category, then so is

A f B 0 g 0 C h D.\array{A & \overset{f}{\to} & B & \to & 0\\ \downarrow && \downarrow^g && \downarrow\\ 0 & \to & C & \underset{-h}{\to} & D.}

This seeming paradox is resolved by noticing that although these two bicartesian diagrams have the same object DD at their lower-right-hand corner, the different maps hh and h-h cause these diagrams to induce different isomorphisms DSAD \cong S A. The isomorphism for the latter diagram incorporates an extra minus sign, relative to the first one, causing these two diagrams to both induce the same triangle AfBgChSA A \overset{f}{\to} B \overset{g}{\to} C \overset{h}{\to} S A in the homotopy category.

References

See derivator for general references about derivators, and also pointed derivator. References particularly pertaining to the stable version include:

  • Alex Heller, “Stable homotopy theories and stabilization” MR

  • Denis-Charles Cisinski and Amnon Neeman, “Additivity for derivator K-theory”, MR

  • Jens Franke, Uniqueness theorems for certain triangulated categories with an Adams spectral sequence, K-theory archive

  • Moritz Groth, Derivators, pointed derivators, and stable derivators (pdf)

Revised on September 24, 2012 14:19:20 by Urs Schreiber (82.169.65.155)