Holmstrom Triangulated category

Gelfand and Manin: Methods book

Neeman book!

See notes by Murfet: Part I, Part II and Part III

Most triangulated cats in nature appear as the homotopy category of a stable model category. For an example where this is not the case, see Muro, with an improvement here

Check the webpage of Schwede.

For Bousfield localization, thick triangulated subcats, and Verdier quotients, see this nLab entry.

For the enriched version of the octahedron axiom (Beilinson, Bernstein, Deligne), see Balmer: Triangular Witt groups. I. The 12-term localization exact sequence (2000)

Balmer and Schlichting: Idempotent completion of triangulated categories.

Beilinson claims on some lecture notes that our notion of triangulated cat is not the right one. Why?

Baues and Muro has a notion of triangulated track category, a 2-dimensional analogue of triangulated categories. In this setting the octahedron axiom holds automatically because of the 2-category structure. See Baues: Triangulated track categories (2006), or ask Muro.

nLab entries

http://ncatlab.org/nlab/show/mapping+cone

http://www.ncatlab.org/nlab/show/triangulated+category

http://www.ncatlab.org/nlab/show/cohomological+functor

Brown representability

There are abstract representability theorems for cohomological functors on triangulated categories. Explain more about this.

Some references

Kuenzer on Heller triangulated cats. See also Schwede’s web page, as well as work by Neeman and by Muro.

Neeman book.

J. P. May, The axioms for triangulated categories, preprint.

A. I. Bondal, M. M. Kapranov, Framed triangulated categories, (Russian) Mat. Sb. 181 (1990), no. 5, 669-683

Weight structures, weight filtrations, weight spectral sequences, and weight complexes (for motives and spectra) , by Mikhail V. Bondarko: http://www.math.uiuc.edu/K-theory/0843 or arXiv

Murfet on cocoverings and wellgeneratedness.

Title: Complete intersections and derived categories. Authors: D.J.Benson, J.P.C.Greenlees. We propose a definition of when a triangulated category should be considered a complete intersection. We show (using work of Avramov and Gulliksen) that for the derived category of a complete local Noetherian commutative ring R, the condition on the derived category D(R) holds precisely when R is a complete intersection in the classical sense. http://arxiv.org/abs/0906.4025

http://ncatlab.org/nlab/show/enhanced+triangulated+category

Notes from Kashiwara-Schapira

An additive category with translation is an additive category together with an additive autofunctor. A morphism of such categories is required to commute (up to isomorphism) with the translations. A triangle in an additive category with translation is a sequence $X \to Y \to Z \to TX$. A morphism of triangles is what you think it is (require last vertical map to be T(first map) ).

Remark: Sign rule for changing signs in the maps of triangle (“The new triangle is IMic to the old if the signs multiply to one”).

A triangulated category is an additive category with translation, together with a family of distinguished triangles satisfying six axioms:

• TR0: “Closed under isomorphism”
• TR1: “X \to X \to 0 \to TX” (first map is id)
• TR2: Every map fits as the first map into some d.t.
• TR3: A triangle X, Y, Z, TX is distinguished iff Y, Z, TX, TY (all maps with a minus sign) is distinguished.
• TR4: A “morphism of triangles but without the third map” can be completed (non-uniquely) to a morphism of triangles.
• TR5: Octahedron axiom

A triangulated functor of triangulated cats is a functor of add cats w transl, sending dt’s to dt’s.

Prop: In a dt, the map $X \to Z$ is zero.

Cohomological functor: A functor from a triangulated category to an abelian category is called cohomological if for any DT (X,Y,Z), the sequence $F(X) \to F(Y) \to F(Z)$ is exact.

Note that a cohomological functor gives a long exact sequence: $\ldots \to F(T^{-1}Z) \to F(X) \to F(Y) \to F(Z) \to F(TX \to \ldots$

Proposotions: Any Hom functor is cohomological in both variables. In a morphism of DTs, if first two maps are IMs, then so os the third. Criterion for uniqueness in TR4. A triangulated functor is exact. If a tr.category admits direct sums, the direct sum of DTs is a DT. Under some hypotheses, the (Kan???) extension of a cohomological functor to a bigger tr.category is also cohomological.

In a tr.category admitting small direct sums, we can define the notion of a system of t-generators. Morally, this is a family of objects that, regarded as covariant functors, detect when an object is isomorphic to zero, and also has a certain countable direct sum property.

Thm: Let D be a triangulated category with a family F of t-generators. Let $H: D^{op} \to Ab$ be a cohomological functor which commutes with small products, i.e. $H( \oplus X_i) \cong \prod H(X_i)$. Then H is representable. Many variants on this thm, for example for cats not admitting small direct sums.

Homotopy colimit. In a tr.category, a hocolim of an inductive system $X_0 \to X_1 \to \ldots$ is defined by the DT:

where the first map is $(id - sh_X)$

CHAPTER 11: Consider an additive category $\mathcal{A}$ with translation. A differential object is a morphism $d_X: X \to TX$. A complex is a DO such that $T(d_X) \circ d_X = 0$. Have natural notion of morphism of DOs, so get categories $\mathcal{A}_d$ and $\mathcal{A}_c$. We define the “shifted object” as $(TX, - T(d_X) )$. Let $f: X \to Y$ be a morphism in $\mathcal{A}$, where $X, Y \in \mathcal{A}_d$. We define the mapping cone of $f$ as $TX \oplus Y$, with differential given by a 2x2 matrix reading $d_{TX}$, $0$, $T(f)$, $d_Y$. Prop: MC(f) will be a complex iff $f$ is a morphism of complexes. MC is a functor from $Mor(\mathcal{A}_d)$ to $\mathcal{A}_d$ which commutes with functors of add.cats with translation. Define the mapping cone triangle to be

where the second map is $0 \oplus id_Y$ and the third is $(id_{TX}, 0)$.

A morphism in $\mathcal{A}_d$ is homotopic to zero if there exists $u: X \to T^{-1}Y$ such that

Note: A functor of add.cats w tr. sends zero-homotopic morphisms to zero-homotopic morphisms.

Now define the homotopy category $K_d(\mathcal{A})$ by taking the objects from $\mathcal{A}_d$, but modding out the Hom groups by the zero-homotopic morphisms. This is also an additive category. w. tr.

Define a DT in the homotopy category to be a triangle IMic to a mapping cone triangle. Thm: $K_d(\mathcal{A})$ is triangulated. The subcategory $K_c(\mathcal{A})$ is a triangulated full subcategory. Prop: Any functor (or bifunctor) of add.cats w.tr. induces triangulated functors on $K_d$ and $K_c$.

Most interesting examples seem to come from complexes in additive categories. Let $C$ be additive. Can form the associated graded category by taking $C^{\mathbb{Z}}$ as objects and translation defined by $(TX)^n = X^{n+1})$. This is an add.category w.tr. From this we form the category of complexes, $Comp(C) = Gr(C)_c$. Notation $X[n] = T^n X$. We have important subcats $Comp^*(C)$ where $*$ is in ${b, +, -, ub, [a,b]}$ (meaning bounded, bounded below, bounded above, unbounded). These are all additive. Both $C$ and $Mor(C)$ can be viewed as subcats of $Comp^b(C)$.

nLab page on Triangulated category