**33**

votes

**3**answers

2k views

### 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 ...

**24**

votes

**2**answers

528 views

### 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 $\mathcal C$ ...

**23**

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 ...

**19**

votes

**2**answers

1k 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 ...

**16**

votes

**0**answers

309 views

### 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 ...

**16**

votes

**0**answers

309 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 ...

**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 ...

**14**

votes

**3**answers

2k views

### 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 ...

**13**

votes

**1**answer

462 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 ...

**13**

votes

**1**answer

574 views

### 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.
...

**12**

votes

**3**answers

1k 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 ...

**12**

votes

**3**answers

905 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 ...

**12**

votes

**5**answers

1k views

### 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 ...

**12**

votes

**2**answers

572 views

### 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) ...

**12**

votes

**0**answers

415 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 ...

**11**

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 ...

**11**

votes

**2**answers

2k 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 ...

**10**

votes

**2**answers

1k views

### 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 ...

**10**

votes

**0**answers

168 views

### 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 ...

**9**

votes

**4**answers

898 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
...

**9**

votes

**3**answers

1k 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 ...

**9**

votes

**4**answers

650 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, ...

**9**

votes

**1**answer

352 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 ...

**9**

votes

**1**answer

361 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, ...

**9**

votes

**1**answer

661 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 ...

**9**

votes

**2**answers

440 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 ...

**9**

votes

**1**answer

774 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 ...

**8**

votes

**4**answers

2k views

### 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 ...

**8**

votes

**3**answers

302 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 & ...

**8**

votes

**1**answer

840 views

### 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 ...

**8**

votes

**1**answer

699 views

### Do homotopy colimits always commute with homotopy colimits?

Do homotopy colimits commute with homotopy colimits? The setting I am thinking of is that of a triangulated category with a model, but it would be interesting to have more general answers as well. A ...

**7**

votes

**2**answers

451 views

### Is the tensorproduct of a triangulated category with a ring again triangulated?

$\underline{Background}$ :
Suppose $\tau$ is a preadditive category and $R$ a ring. Then one may form a new preadditive category $\tau \otimes R$ in the following way:
$\tau \otimes R$ has the same ...

**7**

votes

**3**answers

761 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 ...

**7**

votes

**1**answer

1k 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 ...

**7**

votes

**3**answers

730 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 ...

**7**

votes

**0**answers

223 views

### Do there exist “non-algebraic tensor products” for “algebraic” triangulated categories?

Let us call a triangulated category algebraic if it admits a differential graded enhancement (i.e., an enrichment in complexes of abelian groups). Certainly, there is a notion of a tensor product on ...

**7**

votes

**0**answers

410 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 ...

**7**

votes

**0**answers

190 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 ...

**7**

votes

**0**answers

286 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 ...

**6**

votes

**4**answers

2k views

### 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 ...

**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 ...

**6**

votes

**1**answer

146 views

### Doubt regarding the definition of slice filtration

Voevodsky defined the slice filtration on the motivic stable homotopy category $SH(S)$ over a Noetherian scheme $S$. In the article Open Problems in the Motivic Stable Homotopy Theory, I, Section 2, ...

**6**

votes

**1**answer

800 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 ...

**6**

votes

**0**answers

96 views

### t-structures on the tensor product of stable $\infty$-categories, II

I fork from this thread, a bunch of questions stemmed from a private conversation about that thread. Speculating a bit on the definition of the tensor operation between t-structures generated some ...

**6**

votes

**0**answers

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

**5**

votes

**3**answers

520 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 ...

**5**

votes

**1**answer

132 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 ...

**5**

votes

**1**answer

141 views

### Left orthogonals to compact objects in triangulated categories: existence and “control”?

Let $C$ be a compactly generated triangulated category. Can it contain a non-zero object $M$ such that there are no non-zero morphisms FROM $M$ into compact objects? I would be grateful for any ...

**5**

votes

**1**answer

232 views

### Stable equivalence and triangulated equivalence of self-injective 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

515 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 ...