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.

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The (upper hat of) an octahedral diagram in (la)tex

I would like to draw an octahedral diagram in my paper; I would prefer to present it as the 'upper hat' + the 'lower hat' (as it is common in the texts on triangulated categories). Could anyone tell ...
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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 ...
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427 views

Morphisms between pure complexes of sheaves

I would like to understand the theory of pure complexes of (etale?) sheaves (of geometric origin?). In particular, I would like to understand which conditions are realy necessary in (part 1 of) ...
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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 ...
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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. ...
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Is K(R-Mod) compactly generated when R is an artin algebra?

I wonder if the triangulated category K(R-Mod) is compactly generated when R is an artin algebra? R-Mod denotes all left R-modules. I understand this would be true if R has finite representation type ...
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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 ...
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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 ...
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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 torsion pair in category of modules is as follows: Definition: A pair ...
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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 ...
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Relative Frobenius Structure on the Category of G-modules

Let $G$ be a group $H\leq G$ a subgroup of finite index. Further, let ${\mathcal E}^G_H$ denote the class of those short exact sequences of $G$-modules (over some fixed base ring) which split when ...
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Motivation for Cosuspended Category Axioms

Today I was wondering about the axioms given by Bernhard Keller for Cosuspended Categories. The axioms of a triangle feel very much like exactness, but not quite. The last axiom about the large ...
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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 ...
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Derived Functors in arbitrary triangulated categories

Let ${\mathcal D}$ be a triangulated category, ${\mathcal C}$ a triangulated subcategory and $Q: {\mathcal D}\to {\mathcal D}/{\mathcal C}$ the corresponding Verdier-localization. Now suppose we have ...
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Any relationship of frobenius homomorphism and frobenius category?

I did not understand number theory or characteristic p-algebraic geometry at all. I just know a little about frobenius homomorphism between two schemes. On the other hand, when I learned something on ...
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exactness in triangulated categories is reflected by hom-functor

let $T$ be a triangulated category and $A \to B \to C \to A[1]$ a triangle in $T$ such that for every $A_0 \in T$ the induced long sequence $... \to \hom(A_0,A) \to \hom(A_0,B) \to \hom(A_0,C) \to ...
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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 ...
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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 ...
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Splitting in triangulated categories

Using the axioms for a triangulated category, is it possible to prove the following: $A\stackrel{0}{\to}B\to A\oplus B\to$ is a distinguished triangle. From the first axiom, the map ...
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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 ...
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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 ...
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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 ...
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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 ...
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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) ...