A triangulated category is an additive category equipped with the additional structure of an autoequivalence (called the translation functor) and a class of of triangles satisfying certain axioms.

learn more… | top users | synonyms

1
vote
0answers
44 views

When the tensor product of motives that are not $0$-(homotopy)-connective can be $0$-connective?

For $t$ being the homotopy $t$-structure for Voevodsky effective motivic complexes when $M\otimes N\in DM_{eff}^{t\le -1}$ ensures that either $M$ or $N$ belongs to $DM_{eff}^{t\le -1}$ (so, we use ...
5
votes
0answers
101 views

Meaning of the support property in the definition of Bridgeland stability condition

Let $(Z,\mathcal{P})$ be a Bridgeland stability condition on a triangulated category $\mathcal{C}$. It is said to satisfy the support property if there exists a norm on $K(\mathcal{C})\otimes_\mathbb{...
5
votes
1answer
176 views

Switching left and right adjoints in recollement situations

In Fascieaux pervers, Beilinson, Bernstein and Deligne define a recollement situation as a triple of triangulated categories $\mathcal{D}_U,\mathcal{D}_F$ and $\mathcal{D}$, together with functors $$ ...
16
votes
0answers
325 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 ...
4
votes
1answer
91 views

A distinguished triangle of mapping spectra arising from recollement

I suspect the following should be well known, in some circles, under some name. Alas, I could not figure out how to prove it or where to look it up. Recall that a recollement is a sequence of (...
24
votes
2answers
554 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 $\mathcal C$ ...
6
votes
1answer
150 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, ...
4
votes
1answer
190 views

When may “summand of” be dropped from the definition of perfect dg module?

Let $\mathcal{A}$ be a small dg category. In Section 1 of Lunts-Orlov http://arxiv.org/pdf/0908.4187v5.pdf, $Perf(\mathcal{A})$ is defined to be the full DG subcategory of $\mathcal{S}\mathcal{F}(\...
1
vote
0answers
117 views

What are some standard operations on the set of localizations of a triangulated category?

Let $C$ be a triangulated category, and $D \subset C$ a triangulated subcategory whose inclusion functor admits a right adjoint. Say that a map $x \to y$ in $C$ is a "D-equivalence" if its cone ...
0
votes
0answers
40 views

Equivalence between recollements

Suppose you have a pair of recollements $$ \mathcal{A}' \stackrel{\overset{i^*}{\longleftarrow}}{\stackrel{\overset{i_*}{\longrightarrow}}{\underset{i^!}{\longleftarrow}}} \mathcal{A}\stackrel{\...
2
votes
0answers
47 views

An equivalence between projective modules over the preprojective algebra and an orbit category

Let $Q$ be a Dynkin quiver, and let $kQ$ be its path algebra over some field k. Let $\Pi$ be the preprojective algebra of $Q$. Then (c.f. Section 7.3 of Keller's On Triangulated Orbit Categories) the ...
2
votes
1answer
79 views

Uniqueness of decomposition for a slicing of triangulated category

I am reading Bridgeland's paper on stability condition, where he defined a slicing of a triangulated category. See http://annals.math.princeton.edu/wp-content/uploads/annals-v166-n2-p01.pdf After ...
6
votes
0answers
99 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 ...
3
votes
1answer
188 views

comparison of truncations

I am trying to understand the proof of Lemma 3.0.15 of this paper (Ben-Zvi, Nadler, Preygel - Integral transforms for coherent sheaves). The context is of two triangulated categories $C,D$ with t-...
4
votes
1answer
102 views

Is this additive equivalence a triangulated equivalence?

Let $\mathcal{C}$ and $\mathcal{D}$ be triangulated categories, and suppose that there exists an additive equivalence $F: \mathcal{C} \to \mathcal{D}$. Suppose further that $\mathcal{C} = \text{add}(...
2
votes
1answer
129 views

How do you rigidify a Bousfield localization?

I'm learning about Bousfield localizations. For a triangulated category satisfying some axioms, a Bousfield localizations can be described as an idempotent functor $L:D \to D$. I thought there is a ...
5
votes
0answers
224 views

Do differential objects form triangulated categories?

Let $\mathcal{A}$ be a (fixed) additive category. To a differential object $(A,a)$ for $\mathcal{A}$ (so, $a:A\to A$ and $a^2=0$) one may associate an $\mathcal{A}$-complex $\dots \to A\stackrel{a}{\...
3
votes
1answer
236 views

Reference for t-structures on stable model categories

What kind of definitions of t-structures on stable model categories have been investigated in the literature? Of course, one can always define a t-structure on a stable model category as a t-...
7
votes
0answers
225 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 ...
13
votes
1answer
578 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. ...
5
votes
1answer
288 views

An equivalence of derived categories by Happel-Reiten-Smalø

I have a problem in understanding the proof of a theorem by Happel-Reiten-Smalø. The original reference is this article http://arxiv.org/abs/0911.4473 . I write down the text of the theorem and a ...
5
votes
1answer
235 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 ...
4
votes
1answer
139 views

Interchanging the tensor product with infinite product

Let $R$ be a $k$-algebra (not necessary commutative) and let $\mathbf{D}(R)$ be its derived category (right modules). I'm interested in the class of objects $V$ of $\mathbf{D}(R^{op})$ having the ...
3
votes
0answers
140 views

Can one complete a morphism of commutative triangles to a “commutative cube” in a triangulated category?

This question is a continuation of Can one extend a morphism of commutative triangles to a morphism of octahedral diagrams?. I am deeply grateful for the contributions there; they roughly say that ...
4
votes
0answers
194 views

Can triangulated categories be “approximated by countable subcategories” (that are triangulated but not full!)?

For a given (finite) set of (objects and) morphisms $f_i$ in a triangulated category $C$ I am interested in a (non-full!) triangulated subcategory $C'\subset C$ of "small size" that would contain them....
2
votes
0answers
118 views

Full exceptional collections in derived category of coherent sheaves on non-compact varieties

Let $X$ be a smooth algebraic variety over $\mathbb{C}$, and $D^b(X)$ its bounded derived category of coherent sheaves. Then a full exceptional collection will lead to significant simplifications in ...
4
votes
1answer
309 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 $Mor_{C}(d,e)=\{0\}\ \forall d\in D$; ...
10
votes
0answers
172 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 ...
1
vote
0answers
66 views

Reduced products of (abelian and triangulated) categories: references?

For a filter $U$ on a set $X$ and for a family of categories $C_x$ indexed by $X$ I would like to consider the (corresponding categorical version of) reduced product of $C_x$ (for $x\in X$) with ...
5
votes
1answer
219 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 ...
1
vote
0answers
91 views

$n$-recollements and perverse t-structures

A recent preprint on arXiv brought my attention on the notion of $n$-recollement (def. 2) a generalization of the notion of recollement among three abelian or triangulated categories behaving like a "...
2
votes
0answers
85 views

TTF triples are recollements

The notion of recollement $$ \mathcal{A}' \stackrel{\overset{i^*}{\longleftarrow}}{\stackrel{\overset{i_*}{\longrightarrow}}{\underset{i^!}{\longleftarrow}}} \mathcal{A}\stackrel{\overset{j_!}{...
3
votes
0answers
131 views

When there exists some “cone” of a morphism of (ind-representable) cohomological functors?

I am interested in cohomological functors from a certain small triangulated category $C$ to abelian groups. The question is: given a tranformation $F\to G$ of two functors of this sort, is it ...
3
votes
2answers
255 views

A conservative, non faithful functor between triangulated categories

Suppose that we have: 1) triangulated categories $C,D$, each equipped with a $t$-structure. 2) triangulated functor $F: C \to D$ which is $t$-exact. 3) $F$ reflects isomorphisms, i.e. is conservative....
5
votes
1answer
143 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 ...
4
votes
1answer
130 views

Extension-closed subcategories of triangulated categories as “almost exact” categories

Did anybody study those subcategories of triangulated categories that are closed with respect to "extensions" (in the sense of distinguished triangles; in particular, any such $B$ is additive)? If we ...
33
votes
3answers
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 ...
4
votes
0answers
248 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 ...
4
votes
0answers
187 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 $B$ ...
1
vote
1answer
227 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 ...
3
votes
0answers
90 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
1answer
64 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 ...
3
votes
0answers
253 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
1answer
343 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 ...
1
vote
0answers
84 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
2answers
309 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 $D^b(X,W)=\bigsqcup_{t\in\mathbb{A}^...
1
vote
1answer
148 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 $\mathcal{D}$ ...
5
votes
1answer
226 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 ...
16
votes
0answers
317 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 ...
7
votes
0answers
419 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 ...