A triangulated category is an additive category equipped with the additional structure of an autoequivalence (called the translation functor) and a class of of triangles satisfying certain axioms.

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Cohomological functor from triangulated category

Say we have a cohomological functor F from a triangulated category $C$ to the category $Ab$ of abelian groups, e.g. $F=Hom(x,-)$, where x is an object in $C$. By definition, such a functor transform ...
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Why not define triangulated categories using a mapping cone functor?

Recall that the usual definition of a triangulated category is an additive category equipped with a self equivalence called $[1]$ in which certain diagrams, of the form $X \to Y \to Z \to X[1]$ are ...
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Proof that the homotopy category of a stable $\infty$-category is triangulated

I've been looking at various general strategies for proving that some category is triangulated, and Lurie manages to prove that a huge class of interesting examples of categories that we know about ...
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Does a triangulated category that possesses a subcategory $B$ of generators with no extensions of non-zero degree between them have to be isomorphic to $K^b(B)$?

Suppose that a triangulated category $C$ contains a full additive subcategory $B$ of (strong) generators (i.e. there does not exist a proper strict triangulated subcategory $C'\subset C$ that contains ...
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Neeman's Lemma 1.4.4 (in his book on triangulated categories)

When proving Lemma 1.4.4 in his book on triangulated categories, Neeman asserts that a certain mapping cone splits into a direct sum of three candidate triangles. I'm unable to see why this is so. ...
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On 'graded polarizable' triangulated categories; are there any mixed Galois module analogues known? Also on mixed realizations

There is a construction by Beilinson (in section 3 of "Notes on absolute Hodge cohomology") of the derived category of graded polarizable mixed Hodge complexes; he also proved that this category is ...
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Any examples known of $K^b(B)$ localized by a set of morphisms (i.e. of complexes of length 1)?

I would like to understand the following setting: for an additive $B$ localize $K^b(B)$ by a set of $B$-morphisms (i.e. by a thick triangulated subcategory generated by some set of two-term ...
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Intuition about the triangulation of a homotopy category K(A)

Let $\cal{A}$ be an additive category. Given a morphism of (cochain) complexes $f:X\rightarrow Y$ we can form the mapping cone $C_f$, which is the complex $X[1]\oplus Y$ with differential given by ...
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Glueing triangulated categories

Hello! Given a triangulated category, one can look for semiorthogonal decompositions into (simpler?) triangulated subcategories. I'd like to know if there's a way to attack the opposite problem, ...
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The residue class functor from a Frobenius category to its stable category induces a functor on cube-shaped diagrams - is it dense?

Let $\mathcal{E}$ be a Frobenius category, i.e. an exact category with sufficiently many bijective objects. (Such as e.g. the category of complexes over an additive category.) Let ...
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Showing the category of perfect complexes on a scheme is essentially small

So I've tried to set up a few meetings with professors to talk about this, but I think one of them forgot about it after a big conference, and the other is still on vacation... in the meantime... At ...
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Sequential colim vs sequential hocolim

Suppose we have some homotopical setting in which we can speak of homotopy colimits. The setting I have in mind at the moment is that of a compactly generated triangulated category with a model, but ...
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Do homotopy colimits always commute with homotopy colimits?

Do homotopy colimits commute with homotopy colimits? The setting I am thinking of is that of a triangulated category with a model, but it would be interesting to have more general answers as well. A ...
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If a colimit of distinguished triangles exists, is it also a distinguished triangle?

Consider the following situation in some triangulated category: We are given a collection of distinguished triangles $A_n \to B_n \to C_n \to A_n[1]$ indexed by the natural numbers, together with maps ...
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Pre-triangulated category that isn't triangulated

I've been working through some of the early parts of Neeman's book on triangulated categories, and he mentions that he does not know of a pre-triangulated category that is not triangulated. Is this ...
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t-structures and higher categories?

I'm curious to find out where the viewpoint of higher categories may be useful so here is a somewhat vague question (which may or may not have a reasonable answer). Given a triangulated category, one ...
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A toy example of a tensor triangulated category?

I've been reading Paul Balmer's paper about constructing a "spectrum of prime ideals" on (essentially small) tensor triangulated category in order to then classify thick subcategories. This is all ...
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Is the tensorproduct of a triangulated category with a ring again triangulated?

$\underline{Background}$ : Suppose $\tau$ is a preadditive category and $R$ a ring. Then one may form a new preadditive category $\tau \otimes R$ in the following way: $\tau \otimes R$ has the same ...
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Do exact faithful functors between triangulated categories detect semi-simplicity?

Recall that in a triangulated category, all monomorphisms split (have a retraction). Let $F:C\to D$ be an exact functor between triangulated categories. It is an easy exercise to see that if $F$ is ...
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638 views

The (upper hat of) an octahedral diagram in (la)tex

I would like to draw an octahedral diagram in my paper; I would prefer to present it as the 'upper hat' + the 'lower hat' (as it is common in the texts on triangulated categories). Could anyone tell ...
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Can one extend a morphism of commutative triangles to a morphism of octahedral diagrams?

Consider two (distinct) octahedral diagrams i.e. diagrams mentioned in the octahedron axioms of triangulated categories (with four 'commutative triangular faces' and four 'distinguished triangular ...
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Morphisms between pure complexes of sheaves

I would like to understand the theory of pure complexes of (etale?) sheaves (of geometric origin?). In particular, I would like to understand which conditions are realy necessary in (part 1 of) ...
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How to write down the determinant of a quasi-isomorphism?

This question about the determinant of a perfect complex reminded me of an old question that I had. The construction of the determinant (as in MR1914072 or MR0437541) is a difficult piece of ...
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When is every “solid” perfect complex faithful?

Let $R$ be a noetherian commutative ring. Consider $D^{perf}(R)=K^b(R-proj)$ the category of bounded complexes of finitely generated projective $R$-modules, with maps of complexes up to homotopy. ...
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Is K(R-Mod) compactly generated when R is an artin algebra?

I wonder if the triangulated category K(R-Mod) is compactly generated when R is an artin algebra? R-Mod denotes all left R-modules. I understand this would be true if R has finite representation type ...
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Verdier duality via Brown representability?

Hello, I wonder if the techniques introduced in Neemans paper: "The Grothendieck duality theorem via Bousfield's techniques and Brown representability " can be used to establish Verdier duality. More ...
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Why do people “forget” Verdier abelianization functor?(Looking for application)

I am now learning localization theory for triangulated catgeory(actually, more general (co)suspend category) in a lecture course. I found Verdier abelianization which is equivalent to universal ...
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What is the relationship between t-structure and Torsion pair?

I am away from Torsion theory in abelian category for some while. So it might be a stupid question. The definition of a torsion pair in the category of modules is as follows: Definition: A pair ...
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Tate Cohomology via Stable Categories

Situation Let $G$ be a finite group and provide $G\text{-mod} := {\mathbb Z}G\text{-mod}$ with the Frobenius structure of ${\mathbb Z}$-split short exact sequences. Denote by ...
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Relative Frobenius Structure on the Category of G-modules

Let $G$ be a group $H\leq G$ a subgroup of finite index. Further, let ${\mathcal E}^G_H$ denote the class of those short exact sequences of $G$-modules (over some fixed base ring) which split when ...
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Motivation for Cosuspended Category Axioms

Today I was wondering about the axioms given by Bernhard Keller for Cosuspended Categories. The axioms of a triangle feel very much like exactness, but not quite. The last axiom about the large ...
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Is there triangulated category version of Barr-Beck's theorem?

It is well known that Beck's theorem for Comonad is equivalent to Grothendieck flat descent theory on scheme. There are several version of derived noncommutative geometry. I wonder whether someone ...
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Derived Functors in arbitrary triangulated categories

Let ${\mathcal D}$ be a triangulated category, ${\mathcal C}$ a triangulated subcategory and $Q: {\mathcal D}\to {\mathcal D}/{\mathcal C}$ the corresponding Verdier-localization. Now suppose we have ...
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Any relationship of frobenius homomorphism and frobenius category?

I did not understand number theory or characteristic p-algebraic geometry at all. I just know a little about frobenius homomorphism between two schemes. On the other hand, when I learned something on ...
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exactness in triangulated categories is reflected by hom-functor

let $T$ be a triangulated category and $A \to B \to C \to A[1]$ a triangle in $T$ such that for every $A_0 \in T$ the induced long sequence $... \to \hom(A_0,A) \to \hom(A_0,B) \to \hom(A_0,C) \to ...
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Compact generation for modular representations

Are the derived categories of modular representations of algebraic groups compactly generated? (e.g. consider SL_2 in characteristic 2). Note modular reps of finite groups are compactly generated (by ...
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Categories which are not compactly generated

Do you know natural examples of triangulated categories (or [presentable] stable $\infty$-categories) which are not compactly generated? (ideally they'd be defined algebraically, but curious to hear ...
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Splitting in triangulated categories

Using the axioms for a triangulated category, is it possible to prove the following: $A\stackrel{0}{\to}B\to A\oplus B\to$ is a distinguished triangle. From the first axiom, the map ...
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Classifying triangulated structures on a graded category

I know of several results to the effect that two triangulated categories are equivalent categories (usually one coming from algebra and one coming from topology). However, it's never been clear to me ...
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distinguished triangles and cohomology

Start with A an abelian category and form the derived category D(A). Take a triangle (not necessarily distinguished) and take it's cohomology. We obtain a long sequence (not necessarily exact). If the ...
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Sources for exact triangles in triangulated categories.

The other day I came across the statement that in the triangulated category $\mathfrak{KK}$ (of C*-algebras with KK-groups as morphism sets) "there are many other sources of exact triangles besides ...
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Localization(s) of Categories

I've been trying to read a paper by Krause and came across a strange (to me, of course) notion of localization. After looking around for a long time, and then finding this on his site, I see that ...
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Is there a constructive description of type in the p-local stable homotopy category?

The title pretty much sums it up - but let me give a little bit of background first. In the p-local stable homotopy category (basically one localizes away the torsion spectra which are not p-torsion) ...