**4**

votes

**0**answers

77 views

### On various “extension closures” and “orthogonals” in triangulated categories

A vague form of my question is the following one: for a class of objects $D$ of a triangulated category $C$ we consider the class $E$ of objects that satisfy $Mor_{C}(d,e)=\{0\}\ \forall d\in D$; ...

**4**

votes

**0**answers

154 views

### On a theorem of Hopkins-Neeman-Thomason on generators of thick subcategories of perfect complexes

Notations and background. Let $R$ be a commutative noetherian local ring and let $D(R)$ denote the derived category of the category of R-modules. A strictly perfect complex on $R$ is a bounded complex ...

**1**

vote

**1**answer

191 views

### Does a fully faithful functor between triangulated categories induce embedding of their Grothendieck groups?

Let $\mathcal{A}$, $\mathcal{B}$ be two triangulated categories and $F: \mathcal{A}\to \mathcal{B}$ be a triangulated functor between them. Then $F$ induces an homomorphism between their Grothendieck ...

**2**

votes

**0**answers

71 views

### About some distributive laws in the Bousfield lattice

It is know that for any $\alpha$-well generated tensor triangulated category $\mathcal{T}$ the collection of Bousfield classes forms a set, furthermore this set is a complete lattice, denoted by ...

**1**

vote

**1**answer

58 views

### Quotients of termwise split injections, for additive categories

In the stack exchange notes found in Section 10 of this file, it is claimed that the category $K(\mathcal{A})$ of complexes up to homotopy is a triangulated category, if $\mathcal{A}$ is additive. In ...

**4**

votes

**0**answers

146 views

### 2-periodic derived equivalence

Let $A$ and $B$ be finite-dimensional algebras with finite global dimension over some field (in fact I am thinking of rational incidence algebras of finite posets).
Suppose we know that $A$ and $B$ ...

**2**

votes

**0**answers

229 views

### Functor of points of a tensor triangulated category

Is there is a functor of points approach to tensor triangulated categories parallel to Balmer's theory of prime spectra?
Given a tensor triangulated category $\mathcal{T}$ an $R$-point can be ...

**3**

votes

**0**answers

108 views

### Does the stable category of a nice exact category embed in (the underlying category of) a derivator?

In Derivators, Pointed Derivators, and Stable Derivators, Moritz Groth gives as an example of a non-invertible morphism with trivial cone an inclusion $f:X\to I$. Here $X$ is an object of injective ...

**0**

votes

**0**answers

65 views

### Extending natural transformations of triangulated functors on $D^b(\mathbb P^1)$

Let $\mathcal T = D^b(\mathbb P^1)$, the bounded derived category of $\mathbb P^1$ over a field $k$. It is well known that $\mathcal T$ admits a strong and full exceptional sequence $\{\mathcal O, ...

**1**

vote

**2**answers

231 views

### A question on triangulated category of matrix factorizations

Let $X$ be an algebraic variety and $W:X\rightarrow\mathbb{A}^1$ a regular map, then the triangulated category of matrix factorizations $D^b(X,W)$ is defined to be
...

**1**

vote

**1**answer

90 views

### Is the orthogonal complement of an admissible subcategory admissible itself?

I´m studying Huybrechts book "Fourier-Mukai transforms in algebraic geometry" and I came up with the following: as an example of semi orthogonal decomposition of a triangulated category $\mathcal{D}$ ...

**16**

votes

**0**answers

272 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 ...

**5**

votes

**1**answer

187 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 ...

**4**

votes

**0**answers

314 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 ...

**2**

votes

**0**answers

87 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

339 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

210 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
...

**2**

votes

**0**answers

131 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

143 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 ...

**2**

votes

**0**answers

134 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, $ ...

**4**

votes

**1**answer

216 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

91 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

151 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 $Ind(A^c)$? ...

**1**

vote

**0**answers

45 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 ...

**2**

votes

**1**answer

341 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

**1**answer

85 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

273 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

361 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

332 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 ...

**4**

votes

**3**answers

303 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 ...

**13**

votes

**1**answer

375 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

**2**answers

148 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

177 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

197 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

84 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

224 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 ...

**2**

votes

**0**answers

170 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. ...

**5**

votes

**1**answer

237 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 $A$, ...

**8**

votes

**3**answers

243 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 & ...

**3**

votes

**1**answer

233 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

115 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 ...

**1**

vote

**1**answer

311 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 ...

**2**

votes

**2**answers

229 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 ...

**9**

votes

**1**answer

310 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

**1**answer

230 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

214 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 ...

**5**

votes

**1**answer

119 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

199 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

130 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 ...

**2**

votes

**5**answers

481 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 ...