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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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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Admissibility of intersection of subcategories
Let $\mathscr{T}$ be a triangulated category, and $\mathscr{A}$ be a right admissible subcategory, which means that $i_{\mathscr{A}} : \mathscr{A} \rightarrow \mathscr{T}$ has a right adjoint $i_{\...
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Why are Serre functors always exact?
Let $k$ be a field and $\mathcal{T}$ be a $k$-linear triangulated category with finite dimensional spaces of morphisms. Bondal and Kapranov proved that every Serre functor on $\mathcal{T}$ is exact (...
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Uniqueness of decomposition for a slicing of triangulated category
I am reading Bridgeland's paper on stability condition, where he defined a slicing of a triangulated category. See
http://annals.math.princeton.edu/wp-content/uploads/annals-v166-n2-p01.pdf
After ...
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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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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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Morphism in a Verdier quotient
Let $\mathcal{T}$ be a triangulated category and take $\mathcal{S}$ a triangulated subcategory. Consider the Verdier quotient $\mathcal{T} \left/ \mathcal{S} \right.$, morphisms in this category are ...
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A question on the proof of $D^b(coh(X))\simeq D^b_{coh}(Qcoh(X))$
Proposition 3.5 of "Fourier-Mukai Transforms in Algebraic Geometry" by Huybrechts claims that the is an equivalence of categories
$$
D^b(coh(X))\overset{\sim}{\to} D^b_{coh}(Qcoh(X))
$$
where $D^b(coh(...
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Do we have $D^b_{coh}(X)\simeq D^b(coh(X))$ for a compact complex manifold $X$?
Let $X$ be a compact complex manifold and $\mathcal{O}_X$ be the structure sheaf of holomorphic functions. We call a sheaf of $\mathcal{O}_X$-module $\mathcal{F}$ coherent if it satisfies the ...
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Can homotopy colimits recover cohomology sheaves?
The question is basically the one outlined in the title. Let $\mathcal{T}$ be a triangulated category containing infinite direct sums (e.g. $D_{qc}(X)$ for some separated, finite type over a field $k$,...
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When is the heart of a triangulated category Grothendieck?
Are there conditions which guarantee that the heart of a triangulated category is Grothendieck? Is the compatibility between the t-structure with filtered colimits enough?
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Subcategories of the Verdier quotient?
Let $\mathcal T$ be a triangulated category and $\mathcal C$ a thick triangulated subcategory. We consider the Verdier quotient $\mathcal T/\mathcal C$.
Is there a bijective correspondence between ...
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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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Equivalences between $D^b(\mathcal{B})$ and $\mathcal{D}_{\mathcal{B}}$, the triangulated category generated by $\mathcal{B}$
In the paper "Finite dimensional algebras and highest weight categories" of Cline, Parshall and Scott is stated as follows:
Let $\mathcal{B}$ be an abelian subcategory of a triangulated category $\...
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Example of a tensor triangulated category with two different monoidal t-structures?
What's an example of a tensor triangulated category / symmetric monoidal stable $\infty$-category with two different monoidal $t$-structures?
While I'm at it: is there an example of a tensor ...
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Computing a cone in a $\otimes$-triangulated category
I have a $\otimes$-triangulated category $\mathcal T$ and two triangles in $\mathcal T$:
$$
x_0\to x_1\to c_x\to \Sigma x_0\ \ \ \text{and}\ \ \ y_0\to y_1\to c_y\to \Sigma y_0.
$$
Consider the ...
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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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Reconstructing an isomorphism of exact triangles in the homotopy cat. of a dg-cat. using a functorially induced isomorphism between cones
This question is strictly related to this other one.
Let $\mathcal A$ be a (strongly) pretriangulated dg-category over a field $k$. Let
(source: presheaf.com)
be a diagram in $Z^0(\mathcal A)$, ...
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Homotopy pullbacks and pushouts in stable model categories
There are lots of similar questions that have been answered on this topic (particularly Homotopy limit-colimit diagrams in stable model categories), but I have a specific question that I do not ...
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Which set of compact objects generates the subcategory of a compactly generated stable model category?
I couldn't find any info on what set of compact objects generates the following subcategory:
Let $k$ be a field of positive characteristic and let $G$ be either a finite group or a finite group ...
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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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Are there universal homological functors?
There is a bifunctor $H: Stab^{op} \times Ab \to Top$ where $H(C,A)$ is the space of homological functors $C \to A$. Is this bifunctor left or right representable?
That is, for each small abelian ...
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Abelianization derivator
About ten-fifteen years ago, when the theory of abstract triangulated categories reached a culminating point (after the publication of Neeman's book http://hopf.math.purdue.edu/Neeman/triangulatedcats....
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Filtered triangulated category examples
I am reading Beilison Ginsburg Schechtman's "Koszul duality". In the section 1.3, they introduced the notion filtered triangulated categories with only one example, considering an abelian category ...
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Serre functors for non-proper categories
One usually defines a Serre functor to be a functor on a $k$-linear category $\mathcal{C}$ which has finite dimensional $Hom$s over $k$. In that case, the standard definition is that a Serre functor $...
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Weak generators of the right-bounded derived category of a finite-dimensional algebra
The setup:
Let $A$ be a finite-dimensional $k$-algebra over some field $k$.
Let $\mathcal{B} = Hot^-(Proj \, A)$ denote the homotopy category of cochain complexes of (possibly infinitely generated) ...
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Filtrations of spectra related to cellular ones and singular homology
I would like to study filtrations of spectra (i.e., objects of the "topological" stable homotopy category $SH$; a filtration of a spectrum $E$ is a sequence of compatible maps $E_{\le i}\to E$) whose ...
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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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When is $\Omega^1$ an equivalence?
Let $C$ be an abelian category with enough projectives and $\underline{C}$ the stable category of $C$ that is obtained by factoring out projective modules.
When is the functor $\Omega^1 : \underline{...
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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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Objects of which Grothendieck abelian categories have elements?
The Freyd-Mitchell embedding theorem is a very useful tool for dealing with small abelian categories. However, it does not allow to use "elements" of objects of an abelian category $A$ in those ...
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Highest weight category and weight structures
In various branches of representation theory, there is a notion of highest weight category.
On the other hand there is a notion of weight structure on a triangulated category $C$ (introduced 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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Generators of unbounded derived categories of (quasi-)coherent sheaves
An object $T$ in a triangulated category $\mathcal{D}$ is called a generator if $T^\perp=0$, which means that for any nonzero $X$ in $\mathcal{D}$, there are $i\in\mathbb{Z}$ and a nonzero morphism $T[...
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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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Why do we need the negative sign in TR2 (the "turning triangle" axiom) in the definition of triangulated categories?
In the TR2 of the definition of triangulated categories, we add a negative sign to an arrow when we turn the triangles. What is the significance/motivation of that negative sign ($-u[1]$ as in the ...
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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
$$A ...
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A question on Voevodsky´s categories
I want to try to understand the Voevodsky´s big triangulated categories of motives $DM$ and $DM^{eff}$. Unfortunately, I am being not able to find answers to the following, too vague, questions:
1.- ...
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(Middling) good morphisms of triangles
Neeman in his article "Some new axioms for triangulated categories" calls a morphism of distinguished triangles
$$\require{AMScd}
\begin{CD}
X @>>> Y @>>> Z @>>> X [1] \\
@...
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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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On various relations between "additional axioms" for AB4 and Grothendieck abelian categories
Let $A$ be an abelian category that has a generator and satisfies the AB4 axiom. I would like to understand (better) the relations between various additional "restrictions" on $A$.
So here is my list ...
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Comparing self-equivalences of a triangulated category and automorphisms of its Grothendieck group
There is a homomorphism from the group of (isomorphism classes of) self-equivalences of a triangulated category to the automorphism group of its Grothendieck group. Is this homomorphism surjective? If ...
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Prime spectrum of the derived category of holonomic $\mathcal{D}$-modules?
Let $X$ be a smooth algebraic (/projective if it simplifies things considerably) variety over $\mathbb{C}$ and consider the derived category $\mathcal{C}=D_h^b(\mathcal{D}_X)$ of bounded complexes of $...
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Triangulated categories generated by a collection of submodules
In the book "Cohomology Rings of Finite Groups", by Carlson- Townsley - Elizondo there is the following corollary
The category $\mathsf{stmod}_{\mathbb{k}G}$ is generated as a triangulated category ...
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Generating $K^b(\mathrm{proj})$ as a triangulated category from a full subcategory
Let $K^b(\mathrm{proj}\, A)$ be the bounded homotopy category of chain complexes over $\mathrm{proj}\, A$. In Rickard's paper 'Derived categories and stable equivalence', he defines a tilting complex ...
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Extending a natural transformation using a distinguished triangle
$\require{AMScd}$
Let $\mathcal{T}$ be a triangulated category,
and $\mathcal{S}$ a full subcategory of $\mathcal{T}$ (which is not triangulated).
Let $F, G: \mathcal{T} \to \mathcal{T}$ be two ...
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Techniques for Showing Triviality of K_1 of a Higher Category
Suppose $\cal{C}$ is a small stable $\infty$-category. Then, we have its K-theory spectrum $K(\cal{C})$ that gives us K-theory groups $K_n(\cal{C})$ by taking stable homotopy groups. There are ...
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Grothendieck group of limit of categories
I am in the following situation. I have a stable presentable $\infty$-category $\cal{C}$, and a sequence of full stable subcategories $\dots\subset\cal{C}_{-2}\subset\cal{C}_{-1}\subset\cal{C}_0\...
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Do differential objects form triangulated categories?
Let $\mathcal{A}$ be a (fixed) additive category. To a differential object $(A,a)$ for $\mathcal{A}$ (so, $a:A\to A$ and $a^2=0$) one may associate an $\mathcal{A}$-complex $\dots \to A\stackrel{a}{\...
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Thick subcategories
I hope this question is not too trivial for mathoverfolw.
Let $R$ be a commutative ring (with $0\neq 1$) and $D_{Perf}(R)$ the triangulated category of perfect complexes. Let $C$ be a thick ...