**5**

votes

**1**answer

168 views

### What does an endomorphism in a triangulated category give rise to?

Let $D\xrightarrow[]\varphi D\xrightarrow[]kE\xrightarrow[]j\Sigma D$ be an exact triangle in a triangulated category. I am trying to figure out what structure emerges from this on the base of the ...

**15**

votes

**0**answers

224 views

### 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 ...

**3**

votes

**0**answers

259 views

### 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 ...

**1**

vote

**0**answers

73 views

### Is there any numerical obstruction for all perfect complexes on a scheme being strictly perfect?

Let $X$ be a scheme and $E^{\cdot}$ be a cochain complex of sheaves of $\mathcal{O}_X$-modules.
We call $E^{\cdot}$ a strictly perfect complex if $E^{\cdot}$ is a bounded (in both direction) complex ...

**3**

votes

**1**answer

275 views

### Doing some homological algebra in triangulated categories

It's well known that chain complexes are an abelian category, and in particular we can consider chain complexes of chain complexes, i.e. double complexes. Given a double complex $A^{\bullet\bullet} ...

**2**

votes

**0**answers

184 views

### Towards an enhanced version of homological mirror symmetry for affine varieties

Let $X$ and $X^\vee$ be a mirror pair, homological mirror symmetry relates the symplectic geometry of $X$ to the complex geometry of $X^\vee$ via the equivalence of triangulated categories
...

**11**

votes

**1**answer

339 views

### 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 ...

**2**

votes

**0**answers

100 views

### Do all full exceptional sequences of a triangulated category have the same length?

Let $\mathcal{D}$ be a $k$-linear triangulated category and $\langle E_n,E_{n-1},\ldots, E_0\rangle$ be a sequence of objects of $\mathcal{D}$. We call $\langle E_n,E_{n-1},\ldots, E_0\rangle$ an ...

**1**

vote

**0**answers

111 views

### Thickness of the category of perfect complexes with finite length homology

Let $R$ be a commutative Noetherian local ring and let $D(R)$ be the derived category of $R$-modules. Recall that a chain complex $C_\bullet$ of modules over $R$ is called perfect if it is isomorphic ...

**9**

votes

**2**answers

1k views

### 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 ...

**5**

votes

**1**answer

193 views

### 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 $A$, ...

**2**

votes

**0**answers

113 views

### Perf($\mathscr{A}$) and perfect chain complexes

Suppose $\mathscr{A}$ is a dg category and $ \mathcal{D} (\mathscr{A})$ its associated derived category. For an object in $ a \in \mathscr{A}$ there is associated (right )dg $ \mathscr{A}$ module, $ ...

**3**

votes

**1**answer

192 views

### $t$-structure on modules over highly structured ring spectra

Let $Sp$ be a suitably nice model of a suitably nice category of spectra. By this I mean that $Sp$ is a symmetric monoidal closed stable model category, where we can make sense of homotopy "groups" ...

**3**

votes

**0**answers

82 views

### composition of t-structures “par recollement”

It is a classical result in Beilinson-Bernstein-Deligne's "Faisceaux Pervers" (thm. 1.4.9) that given three triangulated categories $\mathbf{D}_0, \mathbf{D}_1, \mathbf{D}_{01}$ and a recollement ...

**5**

votes

**0**answers

133 views

### Compact objects and ind-objects in triangulated categories

question : let $A$ be triangulated category compactly generated by subcategory $A^c$ of compact objects. Consider category of ind-objects $Ind(A^c)$. Is there relation between $A$ and $Ind(A^c)$? ...

**22**

votes

**3**answers

1k views

### 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 ...

**1**

vote

**1**answer

308 views

### Why does Neeman avoid t-structures?

I have a simple question: the book "Triangulated Categories" by A. Neeman aims to be an exhaustive reference about the whole (basic) theory of triangulated categories. So why there is only a single ...

**1**

vote

**0**answers

42 views

### 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
be a diagram in $Z^0(\mathcal A)$, where the rows are ...

**1**

vote

**1**answer

77 views

### On two notions of 'generators' for a 'large' triangulated category

Let $C$ be a triangulated category that is closed with respect to arbitrary small coproducts; let $D$ be some class of objects of $C$. Then it would be natural to say that $D$ generates $C$ either if
...

**9**

votes

**1**answer

236 views

### 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 ...

**4**

votes

**1**answer

351 views

### Proof without using Yoneda's lemma?

Let $\mathscr{T}$ be atriangulated category.
The third axiom for triangulated categories, namely,
if in the diagram
$$\begin{array} 0X ...

**3**

votes

**1**answer

239 views

### Reference for comparison of heart cohomology with standard cohomology

I'm looking for a reference for the following fact (which I believe to be true and should be easy for people who understand how spectral sequences arise from filtrations).
Let A,B be two hearts of ...

**9**

votes

**1**answer

283 views

### 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, ...

**4**

votes

**3**answers

278 views

### Adams Spectral Sequence for Triangulated Categories

We have the Adams SS with
$$ E_2^{p,q} = Ext^{p,q} _{E^*(E)}([S,E],[S,E]) $$
where $E$ is the Eilenberg-Maclane Spectrum yielding $\mathbb{Z}/p$ coefficients.
I was wondering if there is a SS for ...

**7**

votes

**3**answers

693 views

### 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 ...

**2**

votes

**2**answers

142 views

### When is a triangulated category uniquely determined by its generators?

Notation: Let $\mathcal T$ be a triangulated category, and let $\mathcal E$ be a full subcategory of $\mathcal T$. I write $\langle \mathcal E \rangle$ to indicate the smallest strictly full ...

**1**

vote

**0**answers

119 views

### Bootstrap subcategory abelian?

In the book "K-Theory for Operator Algebras" by Bruce Blackadar, the exercise 23.15.8. on page 246 says:
"Let KKN be the full subcategory of KK with objects in N. Show that KKN is abelian category by ...

**1**

vote

**0**answers

189 views

### Factorization systems in a triangulated/stable category

Google is of little help when questioned about "factorization systems in a triangulated category". Are there informations about them? Are there "natural" (meaning: "obviously defined") OFS/WFS in a ...

**0**

votes

**0**answers

74 views

### when does a “triangulated” functor factor over the homotopy category?

The setup is as follows:
We have the category $C$ of chain complexes over some additive/abelian category and want to pass to the category $K$ of chain complexes modulo homotopy.
So we have an ...

**3**

votes

**1**answer

215 views

### Why should noncommutative CYs be dgas?

For a (commutative) CY manifold of dimension $n$, Serre duality implies that there are ifunctorial isomorphisms
$$\operatorname{Hom}_{D^b(X)}(E,F)\cong\operatorname{Hom}_{D^b(X)}(F,E[n])^*$$
in the ...

**8**

votes

**3**answers

207 views

### 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 & ...

**2**

votes

**0**answers

139 views

### Neeman's homotopy limits in stable $\infty$-categories

Related question (I hoped to turn this into an answer for the user, but in the end it became a question on its own right!): this
In the book
Neeman, Amnon. Triangulated categories. No. 148. ...

**3**

votes

**1**answer

207 views

### Full exceptional set on flag variety

Define a full exceptional set in a triangulated category to be a partially ordered set of objects $\Delta_i$ which generate the category and such that $\text{Ext}^\bullet(\Delta_i, \Delta_j) = 0$ ...

**1**

vote

**0**answers

112 views

### The right (not the left 'Suslin complex' one) adjoint to the embedding of $ DM^{eff} $ into $D(ShvTr) $?

Inside the derived category of Nisnevich sheaves with transfers there is the category $DM^{eff} $ of Voevodsky's effective motivic complexes (actually, Voevodsky only considered bounded above ...

**2**

votes

**5**answers

427 views

### A statement for a triangulated category generated by a subset

Let $D$ be a triangulated category (the triangulated category in my mind is $D^{b}(X)$, that is the derived category of bounded complex of coherent sheaves on a smooth projective variety), $A \subset ...

**1**

vote

**1**answer

299 views

### A $2$-torsion version of the motivic stable homotopy category?

For a field $k$ there exists the motivic stable homotopy $SH(k)$; it is compactly generated. My question: does there exist a 'reasonable' functor $p$ from $SH(k)$ to a certain triangulated category ...

**9**

votes

**4**answers

552 views

### 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, ...

**15**

votes

**4**answers

1k views

### 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 ...

**2**

votes

**2**answers

208 views

### Cool Examples of Localisation in Triangulated Cats Besides the Usual

In the theory of triangulated categories there is a hefty literature on localisation -- the most common example in algebra being (variants of) localising the homotopy category of chain complexes over ...

**4**

votes

**1**answer

218 views

### Is the functor $mod(Tw\mathcal{C})\rightarrow mod(\mathcal{C})$ cohomologically full and faithful?

Let $\mathcal{C}$ be a $c$-unital $A_\infty$-category. If $\mathcal{A}$ is a $c$-unital and triangulated $A_\infty$-category, then there is a $c$-unital $A_\infty$- functor
$$Tw: ...

**3**

votes

**1**answer

191 views

### Bousfield localization before and after taking homotopy

The ncatlab says:
Under suitable conditions it should be true that for $C$ a model category whose homotopy category $\mathrm{Ho}(C)$ is a triangulated category the homotopy category of a left ...

**11**

votes

**3**answers

835 views

### 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 ...

**5**

votes

**1**answer

112 views

### Is this a description of the $\aleph_1$-localizing subcategory generated by a compact generator?

This should be obvious but I'm not seeing it:
The $\mathfrak T$ be a triangulated category with coproducts and with a compact generator $A$ (that is, the functor $\mathfrak T(A,\_)$ preserves ...

**2**

votes

**0**answers

193 views

### How do I find abelian subcategories of periodic triangulated categories?

If $T$ is a triangulated category, then the formalism of $t$-structures gives a way to find abelian subcategories inside. You're supposed to find two strictly full subcategories, $T^{\le 0}, T^{\ge ...

**3**

votes

**0**answers

129 views

### Relationship of additive and triangulated structures in the triangulated cateogry

In the triangulated category $T$, say we have two distinguished triangles: $$X \xrightarrow{f} Y \rightarrow C_f \rightarrow X[1], $$ $$ X \xrightarrow{g} Y \rightarrow C_g \rightarrow X[1]$$ Of ...

**1**

vote

**1**answer

547 views

### Where should one go to learn about triangulated categories?

Lurie's book, higher topos theory describes a new notion of a triangulated category, which is apparently much more natural than the usual definition. Obviously by now a great deal of work has been ...

**2**

votes

**1**answer

251 views

### Generators of Thick Subcategories

Suppose we are given a thick subcategory of the compact objects in the homotopy category of modules over a ring spectrum $R$. Are there conditions we can place on $R$, or on the category (compact) ...

**4**

votes

**1**answer

169 views

### Stable equivalence and triangle stable equivalence of selfinjective algebras

Two (finite-dimensional) $k$-algebras $A$ and $B$ are said to be stable equivalent if their stable module categories $\underline{\rm mod}(A)$ and $\underline{\rm mod}(B)$ are equivalent as $k$-linear ...

**5**

votes

**2**answers

375 views

### Resolutions of unbounded complexes and homotopy (co)limits.

I want to understand once and for all what the resolution of an unbounded complex is. I've been trying to read 'Homotopy limits in triangulated categories' by Marcel Bokstedt and Amnon Neeman and ...

**3**

votes

**0**answers

119 views

### Right Notion of Localizing Subcategory in Quasicategory

Given a stable quasicategory $C$, what are the conditions required of a subcategory for it to descend to a localizing subcategory in $Ho(C)$? Is it enough for it to be reflective? Perhaps exact and ...