Questions tagged [triangulated-categories]
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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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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Replacing triangulated categories with something better
Gelfand and Manin in their 1988 book on homological algebra write that the non-functoriality of cones means that "something is going wrong in the axioms of a triangulated category. Unfortunately at ...
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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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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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Complete the following sequence: point, triangle, octahedron, . . . in a dg-category
Let $\mathcal C$ be a pre-triangulated dg-category (or a stable $\infty$-category, if you wish).
An object $X$ in $\mathcal C$ gives a "point":
$$X$$
A morphism $X\xrightarrow f Y$ in $\mathcal C$ ...
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Is this a model for $K$-theory of a triangulated category?
The recent question Complete the following sequence: point, triangle, octahedron, . . . in a dg-category reminded me of something I wanted to clarify long time ago; most likely this is now well known ...
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Why the stable module category?
Let $R$ be a ring (usually assumed to be Frobenius). The stable module category is what you get when you take the category $\mathsf{Mod}_R$ of $R$-modules, and kill the projective modules. (Of course, ...
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Are functor categories with triangulated codomains themselves triangulated?
I'm fairly confident that the following assertion is true (but I will confess that I did not verify the octahedral axiom yet):
Let $T$ be a triangulated category and $C$ any category (let's say small ...
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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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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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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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Is there a cotangent bundle of a stable $\infty$-category?
Let $C$ be a stable $\infty$-category. Is there any categorical construction $C \mapsto T^* C$, where $T^* C$ is another stable $\infty$-category, that specializes to the following?
When $C$ is the ...
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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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Freyd-Mitchell for triangulated categories?
Is there a nice analog of the Freyd-Mitchell theorem for triangulated categories (potentially with some requirements)? Freyd-Mitchell is the theorem which says that any small abelian category is a ...
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What is Known about the $K$-Theory of Fukaya Categories?
Some Background: In Kontsevich and Soibelman's theory of motivic DT-invariants, one is interested in something like the ``number'' of objects in a 3-Calabi-Yau category $\mathcal{C}$ having a fixed ...
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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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The Octahedral Axiom in group theory
$\require{AMScd}$Here are two results about groups:
(The third isomorphism theorem) Suppose that I have $A \triangleleft B \triangleleft C$ and $A \triangleleft C$. Then $C/B \cong (C/A)/(B/A)$. ...
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Stable homotopy category and the moduli space of formal groups
The usual disclaimer applies: I'm new to all this stuff, so be gentle.
It seems like the spectrum, as defined by Balmer, of the stable homotopy category of finite complexes is something like $M_{FG}$,...
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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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Magic behind idempotent-complete categories a.k.a. why (sometimes) be Karoubian is sexier than be Abelian
It is well know that Karoubian categories (also called idempotent-complete categories) are living between additive and
Abelian categories. While one of the most famous advantages to
work with ...
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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 $\underline{G\text{-mod}}...
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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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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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Is $\mathcal{D} \bigl( \mathrm{QCoh}(\mathfrak{X}) \bigr)$ compactly generated?
An object $E$ in a triangulated category $\mathcal{T}$ with (small) coproducts is called compact if the functor $\mathrm{Hom}_{\mathcal{T}}(E,-)$ commutes with arbitrary coproducts or, equivalently, ...
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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 an (essentially small) tensor triangulated category in order to then classify thick subcategories. This is all ...
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Verdier localization for stable $\infty$-categories
Verdier localization is one of the more intuitive ways to localize a triangulated category, "killing" a suitable class of objects via a functor which is universal with respect to this property.
I ...
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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) ...
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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, i....
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A tensor product for triangulated categories?
Many triangulated categories which show up in mathematics, such as derived categories of various sorts, arise as the homotopy category of a stable $\infty$-category.
Stable $\infty$-categories give ...
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The derived category of integral representations of a Dynkin quiver
Let $Q$ be a Dynkin quiver. Let $\mathbb CQ$ be its complex path algebra. It is defined in a way such that modules over $\mathbb CQ$ are the same as representations of the quiver $Q$. Let's write $\...
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Derived categories of (coherent) sheaves of modules: exceptional images, gluing, and proper descent?
I am interested in the properties of (the derived categories) of various categories of (coherent) sheaves of modules (over varieties). I would like to understand in what extent these properties are ...
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A concrete example of the deficiency of triangulated categories?
There seems to be a general sentiment that triangulated categories are not the "correct" notion to use because mapping cones of morphisms are unique, but only up to non-unique isomorphism.
Does ...
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Is there any significance to Bousfield localization in the non-derived context?
The term "Bousfield localization" of a category $C$ is used in roughly two different ways:
There is a general usage (as in model categories or triangulated categories), which $\infty$-categorically ...
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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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Is the derived category of $A$-dg-modules as a dg-category coincide with the ordinary definition of derived category?
Let $A$ be a unital dg-algebra over a base field $k$. We consider the category of (unbounded) right $A$-dg-modules with morphisms closed degree $0$ maps. We denote this category by dg-mod-$A$. We ...
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Use of derivators to the theory of motives?
This is a rather imprecise question but i think this could become a interesting pool of ideas and comments.
The theory of motives has evolved to a complex field of research the moment Voevodsky (and ...
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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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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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Kontsevich's derived noncommutative geometry and Rosenberg's noncommutative 'spaces'
It appears to me (though I may be wrong) that the common opinion is that the main difference between derived noncommutative geometry and Rosenberg's noncommutative 'spaces' is that Rosenberg's version ...
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Understanding Balmer spectra
$\DeclareMathOperator\Spec{Spec}\newcommand{\perf}{\mathrm{perf}}\DeclareMathOperator\SHC{SHC}$I have just finished reading the paper "The spectrum of prime ideals in tensor triangulated ...
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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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When is the homotopy category of an accessible $\infty$-category accessible?
Let $\mathcal C$ be an accessible $\infty$-category, and let $ho(\mathcal C)$ be its homotopy category. I can think of two "trivial" reasons for $ho(\mathcal C)$ to be accessible:
$ho(\mathcal C) = \...
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Vanishing natural transformation exact triangle
This question is a follow-up to this question I asked some time ago. Let $X$ be a smooth projective variety of dimension $n$ over $\mathbb{C}$. Let $\omega \in H^{n}(X,K_X)$, $\omega \neq 0$. Let
$$A ...
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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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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 'the' homotopy colimit of a sequence of exact triangles an exact triangle?
Let
$$
\begin{array}{rccccl}
A_0&\to& B_0&\to& C_0&\to\\
\downarrow & &\downarrow&&\downarrow\\
A_1&\to& B_1&\to& C_1&\to\\
\downarrow & &...
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If the homotopy category is well-generated, must the $\infty$-category be presentable?
Suppose $\mathcal{C}$ is a stable $\infty$-category whose homotopy category is a well-generated triangulated category in the sense of Neeman's book. Must $\mathcal{C}$ be a presentable $\infty$-...
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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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Any news about equivalences of periodic triangulated or $\infty$-categories?
There is a very old question (October 2009) Equivalence of derived categories which is not Fourier-Mukai which has been bumped by improving links to the literature in one of the answers and attracted ...
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How to show twisted complexes over a DG category is again a DG category?
In Bondal and Kapranov's paper enhanced triangulated categories, a twisted complex over a DG category $A$ is a set $\{(E_i)_{i\in \mathbb Z}, q_{ij}: E_i\to E_j\}$, where $E_i$ are objects in $A$, ...