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Results tagged with homological-algebra
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user 126667
(Co)chain complexes, abelian Categories, (pre)sheaves, (co)homology in various (possibly highly generalized) settings, spectra, derived functors, resolutions, spectral sequences, homotopy categories. Chain complexes in an abelian category form the heart of homological algebra.
11
votes
Accepted
Is tensor product exact in abelian tensor categories with duals?
Yes, because if Z has a dual, then in particular Z ⊗ – has a left adjoint (Z* ⊗ –) and hence commutes with limits (and similarly with colimits, but that's automatic if the category is closed monoidal) …
- 25.2k
3
votes
Terminology: Is there a name for a category with biproducts?
"Category with biproducts" is probably the only standard name, but I'm not really fond of it because (at least in my experience) a more natural characterization of these categories satisfying (1)-(3) …
- 25.2k
4
votes
triangulated vs. dg/A-infinity
One nice thing about the (∞,1)-categorical point of view is that being a stable (∞,1)-category is a property of an (∞,1)-category, and not extra structure. So whatever you have, you can probably find …
- 25.2k
18
votes
Is there an infinity × infinity lemma for abelian categories?
For an $n \times n$ square, the "lemma" follows by applying the spectral sequence of a double complex in two different ways: if we first take homology along columns, then the $E^2$ page is $0$, while …
- 25.2k
6
votes
Kernels and cokernels of multicomplex homomorphisms
If I'm not mistaken, multicomplexes in $\mathcal{A}$ are the same as unbounded cochain complexes in another category constructed from $\mathcal{A}$ as follows:
its objects are $\mathbb{Z}$-graded se …
- 25.2k
27
votes
Why does non-abelian group cohomology exist?
To elaborate on Eric's answer, I believe that $H^{1-n}(G, A)$ is $\pi_n$ of the homotopy fixed point space $K(A, 1)^{hG}$. That exact sequence which ends at $H^1$ - which is only a set, while $H^0$ i …
- 25.2k
11
votes
1
answer
940
views
Is ΩΣ in {simplicial commutative monoids} group completion?
Let C be the model category of simplicial commutative monoids (with underlying weak equivalences and fibrations), or equivalently the (∞,1)-category PΣ(Top), where T is the Lawvere theory for commutat …
- 25.2k
24
votes
Why are spectral sequences so ubiquitous?
Let me first try to answer a simpler question:
Why are long exact sequences so ubiquitous?
Almost anything that is written as a capital letter, followed by a subscript i or superscript -i, i an …
- 25.2k
10
votes
Sheaf cohomology and injective resolutions
If you're willing to take for granted that (bounded-below) chain complexes and quasi-isomorphisms are good things to study, then left exact functors have the defect that they do not preserve quasi-iso …
- 25.2k
5
votes
Homological algebra for commutative monoids?
I agree with Charles and Eric that the natural setting for your question is the model category of simplicial commutative monoids. However, my earlier guess that the resulting homotopy theory would ad …
- 25.2k
16
votes
Accepted
How to think about model categories?
Model categories are 1-categorical presentations of (∞,1)-categories, which you can just think of as categories enriched in topological spaces, such as the category of spaces itself. (Actually, there …
- 25.2k