# Questions tagged [triangulated-categories]

For questions about triangulated categories. A triangulated category is an additive category with an additive auto-equivalence called a translation (or shift) functor, and a class of distinguished triangles satisfying various axioms.

196 questions
Filter by
Sorted by
Tagged with
155 views

### Is the octahedral axiom really equivalent to the 4 x 4 lemma?

$\require{AMScd}$I make reference to this paper. I've recently become aware that there's an annoying variety of a priori distinct (but "known" to be equivalent up to adding other hypotheses) ...
• 40.4k
48 views

### tensor product of a graded vector space and an object in k-linear category

In the book "Fourier-Mukai Transforms in Algebraic Geometry", the author has been using the following terminology quite a few times in the first two chapters (Proof of Lemma 1.58, Definition ...
• 1,280
25 views

### Different definitions of sequential homotopy colimits

There are two notions of homotopy limits, one for triangulated categories and one for model categories and I wonder whether these two coincide. More concretely, let $\mathcal{T}$ be a triangulated ...
• 2,109
17 views

### Left adjoint of an additive functor between triangulated categories that commute with shift

Let $\mathcal S, \mathcal T$ be triangulated categories and $R: \mathcal S \to \mathcal T$ be an additive functor that commutes with shift. If $L:\mathcal T \to \mathcal S$ is a left adjoint to $R$, ...
• 525
34 views

### Checking the multiplicative system definition for a class of maps in a triangulated category

I am reading these notes Derived categories, resolutions, and Brown representability Henning Krause (https://arxiv.org/pdf/math/0511047.pdf) about derived and triangulated categories I am having ...
• 617
94 views

### On the two definitions of derived functor in general triangulated category.

I'm learning homological algebra using several references books. But I find two definitions of derived functor in general triangulated category. I wonder to know which definition is more generally ...
• 492
1 vote
65 views

### Basic questions of triangulated functors

I am not familiar with triangulated categories so these questions might be too basic (but I did not find any answers by google). Also, the question can be formulated in purely triangulated category ...
• 39
1 vote
56 views

### Being a triangulated functor is a property or an additional structure?

Let $\mathcal T$ and $\mathcal T'$ be triangulated categories and consider an additive functor $F:\mathcal T \to \mathcal T'$. Some authors say $F$ is triangulated if there exists a natural ...
• 400
51 views

### Dual statement in triangulated category

I want to show that if $X \overset{f}{\rightarrow} Y \overset{g}{\rightarrow} Z \overset{h}{\rightarrow} X[1]$ and $h = 0$ then $f$ is a split monomorphism and $g$ is a split epimorphism. I have ...
• 409
1 vote
58 views

### Distinguished triangles formed from fiber product of a morphism

Let $\mathcal{C}$ be a triangulated category and $f:A\to B$ be morphism with fiber product $F\to A$. I have heard that this leads to a distinguished triangle of the form $$F\to A\to B\to\Sigma F$$ but ...
• 3,412
1 vote
53 views

### When does a morphism of bounded complexes induce distinguished triangles in the bounded derived category?

I have a very basic question about the triangulated structure of bounded derived category of finitely generated modules. Let $R$ be a commutative Noetherian ring, and $\text{mod } R$ be the abelian ...
• 525
40 views

### A question on isomorphism of objects in triangulated categories

I am learning the basics of general triangulated categories, and I have the following question: Let $\mathcal T$ be a triangulated category and let $A,B,C,D$ be objects such that we have two exact ...
• 289
1 vote
91 views

### What is a 'cofiber sequence' in a triangulated category?

I am reading the definition of a localizing subcategory $\mathcal{D}$ of a triangulated category $\mathcal{C}$ (Definition 1.1.1 of the book Axiomatic Stable Homotopy Theory). It refers to a 'cofiber ...
• 3,412
1 vote
96 views

### Compact generators of $D^b_c(X)$

Let $X$ be a scheme and $n$ be an integer invertible on $X$, consider the category $D^b(X,\mathbb{Z}/n)$ (resp. $D^b_c(X,\mathbb{Z}/n)$) of bounded chain complexes (up to quasi-isomorphisms) of ...
• 2,109
122 views

### What is extension closure in triangulated categories?

The term extension closure appears in some papers constructing t-structures on triangulated categories, for example in Section 1.2 of Bayer, Arend; Macrì, Emanuele; Toda, Yukinobu, Bridgeland ...
• 1,373
45 views

• 89
60 views

### Recommendation: Textbooks on representation theory of algebras emphasizing the usage of triangulated categories?

There exist some nice textbooks on representation theory of algebras (This book for example), which mainly develop the theory on module categories (or their quotients). As I'm also learning ...
• 487
1 vote
113 views

### Product, coproduct, and convolution of chain complexes

I am currently reading Gelfand and Manin's Methods of Homological Algebra, and at some point (IV. 10. Exercise 2 I guess? The numbering in the book is quite inconvenient to get around, so I'm ...
• 2,194
1 vote
86 views

### Triangulated Category: Turning the Triangles

I'm trying to learn Triangulated Category by reading some online notes and I have a question about the "turning the Triangles" Axiom. Below is the Axiom from the notes Now, given a ...
• 832
1 vote
99 views

### Isomorphism between distinguished triangles in a triangulated category

$\DeclareMathOperator{\id}{id}$ Throughout this post "d.t." stands for "distinguished triangles". I'm studying some category theory and the book I'm following has the following ...
• 1,048
1 vote
144 views

• 289
104 views

### Syzygy in projective resolution and thick closure

Let $R$ be a Commutative Noetherian ring, and let $\text{mod } R$ denote the abelian category of finitely generated $R$-module. Consider the bounded derived category $D^b(\text{mod } R)$ which is a ...
• 105
1 vote
36 views

### Construction of truncations in a $t$-structure

I'm reading about $t$-structures in the book D-Modules, Perverse Sheaves, and Representation Theory by Hotta, Takeuchi, and Taniski. Given a $t$-structure $(\mathcal{D}^{\leq 0}, \mathcal{D}^{\geq 0})$...
• 739
149 views

### Rotation axiom for the triangulation on a derivator

I'm having some trouble following an argument in Moritz Groth's paper on Derivators, pointed derivators and stable derivators. More precisely, I'm currently stuck on the rotation axiom of the ...
• 8,051
97 views

• 1,632
1 vote
56 views

• 7,295
1 vote
56 views

### Canonical morphism in Huybrecht's Fourier-Mukai Transformations

Let $(\mathcal{D},T)$ be a $k$-linear triangulated category and $A, E \in \mathcal{D}$. Define as $A[i]:= T^i(A)$ the $i$-th twist under endomorphism $T: \mathcal{D} \to \mathcal{D}$. In the proof of ...
• 7,295