3
votes
1answer
167 views

Bousfield localization before and after taking homotopy

The ncatlab says: Under suitable conditions it should be true that for $C$ a model category whose homotopy category $\mathrm{Ho}(C)$ is a triangulated category the homotopy category of a left ...
3
votes
0answers
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 ...
1
vote
0answers
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, ...
7
votes
4answers
666 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 ...
7
votes
0answers
257 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 ...
3
votes
3answers
323 views

Sequential colim vs sequential hocolim

Suppose we have some homotopical setting in which we can speak of homotopy colimits. The setting I have in mind at the moment is that of a compactly generated triangulated category with a model, but ...
6
votes
1answer
605 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 ...
5
votes
2answers
575 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 ...
8
votes
4answers
1k 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 ...
12
votes
2answers
455 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) ...