**3**

votes

**1**answer

194 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

143 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 & ...

**1**

vote

**0**answers

83 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. ...

**4**

votes

**0**answers

78 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 $A$, ...

**3**

votes

**1**answer

176 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

109 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

370 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

275 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 ...

**6**

votes

**4**answers

506 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 ...

**3**

votes

**2**answers

182 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 ...

**5**

votes

**0**answers

170 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

193 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

167 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

793 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

107 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

187 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

123 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

536 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

220 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

154 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

336 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

114 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 ...

**3**

votes

**0**answers

289 views

### Good morphisms of distinguished triangles: can Neeman's method be applied to the motivic stable homotopy category?

It is well known that non-uniqueness of a cone for a morphism in a triangulated category $C$ makes constructing exact functors (of triangulated categories) a difficult task. In section 3 of his "Some ...

**1**

vote

**0**answers

164 views

### The homotopy colimit of a tower of triangles

Set the framework to be a triangulated category with all set indexed coproducts.
In "Relative Homological Algebra and Purity in Triangulated Categories", J. of Algebra 227, (2000), pp. 268- 361, ...

**3**

votes

**1**answer

229 views

### Is the stable homotopy category idempotent complete?

Is the stable homotopy category idempotent complete? I have not been able to prove it, and the proof for abelian groups seems to strongly rely on looking at elements.
Thanks,
Jon

**2**

votes

**1**answer

282 views

### Showing morphism of sheaves is zero

I work in derived category $D^b(X)$ of constructible sheaves on a reasonable space $X$. Let $j\colon U\to X$ be an open inclusion and $i\colon Y\to X$ the closed complement. Let $M,N\in D^b(X)$ and ...

**19**

votes

**1**answer

889 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 ...

**6**

votes

**0**answers

150 views

### Enriched triangulated categories

If $V$ is a monoidal model category, then its homotopy category $\mathrm{Ho}(V)$ is a monoidal category. Similarly, if $M$ is a $V$-model category, then $\mathrm{Ho}(M)$ is a ...

**0**

votes

**0**answers

529 views

### 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. ...

**10**

votes

**0**answers

350 views

### 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 ...

**0**

votes

**1**answer

604 views

### Terminology - subcategories of Abelian categories

Hello,
I have terminological question. Consider the following properties of a full subcategory $B \subset A$, where $A$ is an abelian category, and we assume $B$ to be closed under finite direct ...

**12**

votes

**3**answers

953 views

### 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 ...

**2**

votes

**0**answers

56 views

### Computing morphisms in localizations of $K(B)$

Let $B$ be an additive category (a small one; one can assume that it is a $\mathbb{Q}$-category, yet not much else is known about it). Given a set of objects $S$ of $K^b(B)$ (or $K(B)$), I consider ...

**2**

votes

**1**answer

882 views

### Where could I publish an average paper on triangulated categories?

I have a rather abstract paper on triangulated categories; I would say that it is of average size and quality. I want to find an appropriate journal to publish it; I would like it to be accepted in ...

**9**

votes

**3**answers

978 views

### 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 ...

**8**

votes

**2**answers

1k views

### 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 ...

**8**

votes

**1**answer

631 views

### 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 ...

**6**

votes

**3**answers

1k views

### 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 ...

**8**

votes

**1**answer

520 views

### 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 ...

**15**

votes

**2**answers

806 views

### 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 ...

**7**

votes

**1**answer

996 views

### 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 ...

**9**

votes

**2**answers

411 views

### 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 ...

**2**

votes

**0**answers

129 views

### 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 ...

**1**

vote

**0**answers

39 views

### 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 ...

**7**

votes

**4**answers

664 views

### 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
...

**7**

votes

**0**answers

256 views

### 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 ...

**7**

votes

**3**answers

672 views

### 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 ...

**5**

votes

**3**answers

450 views

### 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 ...

**6**

votes

**1**answer

745 views

### 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 ...