Stable Derivators: Structures & Applications

• Stable derivators are pointed derivators where cartesian and cocartesian squares coincide, ensuring fibers and cofibers agree up to canonical isomorphism.
• They enhance triangulated categories by equipping each diagram category with a canonical triangulated structure and exact morphisms.
• Applications include stabilizing homotopy theories, constructing t-structures, and providing filtered enhancements in both algebraic and topological settings.

A stable derivator is a pointed derivator in which cartesian and cocartesian squares coincide in $D([1]\times[1])$; equivalently, fibers and cofibers agree up to canonical isomorphism, the suspension $\Sigma$ and loop $\Omega$ are inverse equivalences, and homotopy pushouts coincide with homotopy pullbacks (Groth, 2016). Stable derivators provide an enhancement of triangulated categories: for strong stable derivators, each diagram category $D(A)$ carries a canonical triangulated structure, and restriction, Kan extension, and exact morphism functors acquire canonical exact structures (Groth, 2011).

1. Axiomatic framework

A prederivator is a strict $2$-functor

$D:\mathrm{Cat}^{op}\to \mathrm{CAT}.$

A derivator is a prederivator satisfying the standard axioms $(\mathrm{Der1})$–$(\mathrm{Der4})$: $D$ sends coproducts to products, isomorphisms are detected pointwise, and for every functor $u:A\to B$ the restriction $\Sigma$0 admits both adjoints $\Sigma$1 and $\Sigma$2, with pointwise formulas for Kan extensions computed via comma categories (Groth, 2016). Concretely,

$\Sigma$3

where $\Sigma$4 and $\Sigma$5 are the evident projections (Groth, 2016).

A derivator is pointed if $\Sigma$6 has a zero object. In a pointed derivator, every level $\Sigma$7 is pointed and $\Sigma$8 are pointed functors (Coley, 2018). Groth’s simplification shows that pointedness in this sense is equivalent to the stronger formulation in which extension along sieves and cosieves admits extra adjoints (Groth, 2011).

Strongness is an additional condition: for every $\Sigma$9, the partial underlying diagram functor

$\Omega$0

is full and essentially surjective (Lagkas-Nikolos, 2016). Strongness is not part of stability itself, but it is used in canonical triangulation results and in constructions such as filtered enhancements (Groth, 2011).

2. Stability, fibers, cofibers, and canonical triangles

Let $\Omega$1 and $\Omega$2. A pointed derivator $\Omega$3 is stable if cartesian and cocartesian squares coincide in $\Omega$4 (Groth, 2016). Equivalent formulations include: $\Omega$5 is an adjoint equivalence, $\Omega$6 is an adjoint equivalence, cofiber squares are precisely fiber squares, and strongly cocartesian $\Omega$7-cubes are precisely strongly cartesian ones for $\Omega$8 (Groth, 2016).

The canonical suspension and loop are defined by Kan extensions over the square and its corners: $\Omega$9

$D(A)$0

In a stable derivator, $D(A)$1 and $D(A)$2 are inverse equivalences (Groth, 2016). The cofiber functor is

$D(A)$3

and the fiber functor is defined dually via $D(A)$4 and right Kan extension (Groth, 2016).

For a morphism $D(A)$5, the canonical cofiber triangle is

$D(A)$6

Dually, fiber sequences give triangles

$D(A)$7

In the stable case, $D(A)$8 canonically, and the triangulated shift is the derivator suspension (Groth, 2016).

Stable derivators also admit Mayer–Vietoris triangles. For a cocartesian square

$D(A)$9

with vertical maps $2$0 and $2$1, there is a cocartesian square

$2$2

and hence a distinguished triangle

$2$3

(Groth et al., 2013). This supplies the Mayer–Vietoris sequences familiar from stable model categories and stable $2$4-categories.

3. Canonical triangulations and exactness

For a strong, stable derivator $2$5, each $2$6 carries a canonical triangulated structure (Groth, 2016). The shift functor on $2$7 is induced pointwise by the derivator suspension, and distinguished triangles arise canonically from coherent cofiber sequences

$2$8

in $2$9 (Groth, 2016). Groth proved that the values of a stable derivator can be canonically endowed with the structure of a triangulated category, and that the functors belonging to the stable derivator can be turned into exact functors with respect to these triangulated structures (Groth, 2011).

For a morphism of derivators $D:\mathrm{Cat}^{op}\to \mathrm{CAT}.$0, one says that $D:\mathrm{Cat}^{op}\to \mathrm{CAT}.$1 preserves pushouts if it preserves colimits of shape $D:\mathrm{Cat}^{op}\to \mathrm{CAT}.$2, preserves pullbacks if it preserves limits of shape $D:\mathrm{Cat}^{op}\to \mathrm{CAT}.$3, is right exact if it preserves initial objects and pushouts, is left exact if it preserves terminal objects and pullbacks, and is exact if it is both right exact and left exact (Groth, 2016). In stable derivators, exactness collapses: a morphism is left exact iff right exact iff exact (Groth, 2016).

Right exact morphisms preserve finite coproducts, suspension, cones, cofibers, cofiber squares, cofiber sequences, and iterated cofiber sequences; by duality, left exact morphisms preserve finite products, loops, fibers, and related constructions (Groth, 2016). The main exactness theorem states that if $D:\mathrm{Cat}^{op}\to \mathrm{CAT}.$4 is an exact morphism of strong, stable derivators and $D:\mathrm{Cat}^{op}\to \mathrm{CAT}.$5 is any small category, then there is a canonical exact structure

$D:\mathrm{Cat}^{op}\to \mathrm{CAT}.$6

which is a natural isomorphism, and $D:\mathrm{Cat}^{op}\to \mathrm{CAT}.$7 is exact in Verdier’s sense (Groth, 2016). Explicitly, if

$D:\mathrm{Cat}^{op}\to \mathrm{CAT}.$8

is distinguished in $D:\mathrm{Cat}^{op}\to \mathrm{CAT}.$9, then

$(\mathrm{Der1})$0

is distinguished in $(\mathrm{Der1})$1 (Groth, 2016).

The canonicity statement is equally strong. If one chooses different canonical triangulations on $(\mathrm{Der1})$2, the identity functor $(\mathrm{Der1})$3 can be equipped with a canonical exact structure making it an exact isomorphism between the two triangulated structures; in this sense, the triangulations are canonical (Groth, 2016).

4. Characterizations of stability and the Kan-extension calculus

The definition of stability admits several exact reformulations. A pointed derivator is stable if and only if the adjunction $(\mathrm{Der1})$4 is an equivalence, if and only if the adjunction $(\mathrm{Der1})$5 is an equivalence, if and only if cocartesian squares are cartesian, and if and only if the derivator is cofiber-stable or $(\mathrm{Der1})$6-stable (Groth et al., 2013). A further reformulation is that a derivator is stable if and only if it admits a zero object and partial cone and partial fiber morphisms commute on squares (Groth, 2016). Concretely, for every square $(\mathrm{Der1})$7,

$(\mathrm{Der1})$8

is an isomorphism precisely in the stable case (Groth, 2016).

A major characterization is finitary exactness: a derivator is stable if and only if homotopy finite limits and homotopy finite colimits commute (Groth, 2016). Precisely, for homotopy finite categories $(\mathrm{Der1})$9 and $(\mathrm{Der4})$0 and any $(\mathrm{Der4})$1, the canonical comparison

$(\mathrm{Der4})$2

is an isomorphism (Groth, 2016). This theorem was generalized into a relative notion of stability: a derivator is stable if and only if homotopy finite limit functors have right adjoints, and if and only if homotopy finite colimit functors have left adjoints (Groth et al., 2017).

The paper “Revisiting the canonicity of canonical triangulations” gives a systematic analysis of how morphisms of derivators interact with limits, colimits, and Kan extensions (Groth, 2016). For a morphism $(\mathrm{Der4})$3 and a functor $(\mathrm{Der4})$4, the canonical mate for preservation of left Kan extensions is

$(\mathrm{Der4})$5

and dually for right Kan extensions

$(\mathrm{Der4})$6

This leads to the preservation classes $(\mathrm{Der4})$7 and $(\mathrm{Der4})$8, together with closure properties under equivalences, left adjoints, composition, natural isomorphisms, and cancellation for fully faithful functors (Groth, 2016). Preservation of colimits of shape $(\mathrm{Der4})$9 is equivalent to preservation of colimiting cocones in $D$0, and right exact morphisms preserve left homotopy finite left Kan extensions; in stable derivators, exact morphisms preserve both left and right homotopy finite Kan extensions and extensions by zero (Groth, 2016).

These exact structures organize $D$1 into a genuine $D$2-categorical enhancement. Every strong, stable derivator admits a lift to the $D$3-category $D$4 of triangulated categories, exact functors, and exact natural transformations: $D$5 Restriction functors $D$6, and similarly $D$7, acquire canonical exact structures, and natural transformations $D$8 induce exact transformations $D$9 (Groth, 2016).

5. Stabilization, $u:A\to B$0-structures, and algebraic constructions

Stabilization is the universal passage from a pointed homotopy theory to a stable one. For a regular pointed derivator $u:A\to B$1, the derivator of prespectra is

$u:A\to B$2

and an object $u:A\to B$3 is an $u:A\to B$4-spectrum if the canonical map

$u:A\to B$5

is an isomorphism (Coley, 2018). The full subprederivator $u:A\to B$6 of $u:A\to B$7-spectra is a localization of $u:A\to B$8, is stable, and the stabilization morphism is

$u:A\to B$9

(Coley, 2018). Its universal property is

$\Sigma$00

for stable $\Sigma$01, and when $\Sigma$02 is already stable, $\Sigma$03 is an equivalence (Coley, 2018). A common misconception is that strongness is necessary for stabilization; Coley’s revision removes Heller’s strongness assumption and works under regular pointedness (Coley, 2018).

Stable derivators also organize $\Sigma$04-structures uniformly across all shapes. If $\Sigma$05 is strong and stable and $\Sigma$06 is a $\Sigma$07-structure on $\Sigma$08, then for each small $\Sigma$09,

$\Sigma$10

defines a lifted $\Sigma$11-structure on $\Sigma$12, and the heart satisfies

$\Sigma$13

via the diagram functor (Saorín et al., 2017). If the $\Sigma$14-structure is compactly generated, then the coaisle is closed under directed homotopy colimits, and the heart is AB5; in well generated algebraic or topological settings, the heart of any accessibly embedded $\Sigma$15-structure has a generator, hence compactly generated hearts are Grothendieck abelian (Saorín et al., 2017).

Loregian–Virili establish another structural bridge: for a stable derivator $\Sigma$16, suitable derivator factorization systems, namely normal derivator torsion theories, correspond bijectively to $\Sigma$17-structures on the base $\Sigma$18 (Loregian et al., 2017). At the triangulated level, this is the statement that normal triangulated torsion theories correspond bijectively to $\Sigma$19-structures.

Realization functors supply a derivator-level Morita theory. Given a strong stable derivator $\Sigma$20, a $\Sigma$21-structure $\Sigma$22 on $\Sigma$23, and heart $\Sigma$24, one can construct under mild assumptions a morphism of prederivators

$\Sigma$25

If $\Sigma$26 is induced by a suitably bounded tilting or cotilting object, then $\Sigma$27 is an equivalence (Virili, 2018).

6. Examples, applications, and scope

Standard examples of stable derivators include derivators of unbounded chain complexes in a Grothendieck abelian category, spectra in topology, and homotopy derivators of stable model categories, stable cofibration categories, and stable $\Sigma$28-categories (Groth, 2016). The derivator of a dg-category gives a recent dg-enriched construction: for a dg-category $\Sigma$29,

$\Sigma$30

and if $\Sigma$31 is homotopically complete and cocomplete, $\Sigma$32 is a strong, stable derivator on all of $\Sigma$33; for pretriangulated $\Sigma$34, restricting to finite direct categories still yields a strong, stable derivator (Genovese et al., 4 Aug 2025). In the Frobenius case, these values admit direct descriptions via acyclic complexes of projectives or injectives and via Gorenstein projective or Gorenstein injective diagrams (Genovese et al., 4 Aug 2025).

Closed monoidal stable derivators provide a context in which the compatibilities needed for additivity of traces are automatic. In a closed symmetric monoidal stable derivator, if $\Sigma$35 is bicartesian and $\Sigma$36 is a morphism of squares, then for dualizable $\Sigma$37 one has

$\Sigma$38

and in particular

$\Sigma$39

(Groth et al., 2012). This replaces May’s extra axioms for monoidal triangulated categories by derivator-level universal properties.

The Eilenberg–Moore theory of monads also behaves well in the stable setting. If $\Sigma$40 is a cocontinuous monad on a stable derivator $\Sigma$41, then the levelwise module construction $\Sigma$42 is again a stable derivator; if, moreover, $\Sigma$43 is stable, strong, idempotent-complete and $\Sigma$44 is separable, then $\Sigma$45 is strong and stable, and the free and forgetful morphisms are exact (Lagkas-Nikolos, 2016). This produces examples of derivators that satisfy all the axioms for stability except the strongness one, and shows that separability restores strongness under mild additional assumptions (Lagkas-Nikolos, 2016).

Stable derivators also underlie filtered enhancements of triangulated categories. If $\Sigma$46 is a stable derivator, then the underlying triangulated category $\Sigma$47 has an $\Sigma$48-enhancement satisfying the additional axiom $\Sigma$49 (Modoi, 2017). This confirms Bondarko’s conjectural filtered enhancement for triangulated categories arising as underlying categories of stable derivators (Modoi, 2017).

A further application is a derivator-theoretic bounded $\Sigma$50-Dold–Kan correspondence. For any stable derivator $\Sigma$51 and integer $\Sigma$52, there is an equivalence of stable derivators

$\Sigma$53

natural with respect to exact morphisms (Sava, 2022). The construction is purely in the $\Sigma$54-category of derivators and is independent of coefficients; it can also be realized as an action of a spectral bimodule in universal tilting theory (Sava, 2022).

A final scope condition is worth stating explicitly. Represented derivators $\Sigma$55 from complete, cocomplete, pointed categories $\Sigma$56 are pointed but not stable unless $\Sigma$57 is trivial (Groth, 2016). This is one of the clearest indications that stability is not a formal property of ordinary category theory alone: it is a homotopy-theoretic exactness condition encoded by the derivator calculus of Kan extensions, homotopy exact squares, and coherent diagram categories (Groth, 2016).