**17**

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**0**answers

625 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 ...

**14**

votes

**0**answers

456 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.
...

**10**

votes

**0**answers

349 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 ...

**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 ...

**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 ...

**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

**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

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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 ...

**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 ...

**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 ...

**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

**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

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. ...

**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 ...

**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, ...

**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 ...

**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. ...