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Results tagged with homological-algebra
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user 402
(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.
2
votes
Accepted
On the ordered set of real numbers, does sheaf+cosheaf imply constant?
Here is one possible way to proceed.
(Caveat lector: I haven't checked all the details carefully.)
Denote by S the sphere spectrum and by N the (contractible) spectrum that implements a nullhomotopy …
- 37.8k
1
vote
Projective/injective object in functor category
For projective objects, see here: Necessary conditions for cofibrancy in global projective model structure on simplicial presheaves.
As explained there for presheaves of sets (and the same argument w …
- 37.8k
4
votes
Accepted
Proof of derived tensor-hom adjunction
Does the 'derived adjunction' still hold? If yes, how do you prove it? I would prefer a constructive proof which allows me to understand the map.
Yes. The easiest way to get the derived adjunction …
- 37.8k
4
votes
Two equivalent definitions of differential graded algebras
The definition as a sequence of components immediately generalizes to settings like arbitrary abelian categories without countable coproducts.
The definition with all components fused together cannot …
- 37.8k
6
votes
Accepted
Why should we study the total complex?
Chain complexes arise from simplicial abelian groups via the Dold–Kan correspondence: the normalized chain complex functor establishes an equivalence of categories from simplicial abelian groups to (n …
- 37.8k
9
votes
Accepted
Is the adjunction between spaces and chain complexes monadic?
This answer assumes that $\def\Ch{{\sf Ch}}\def\Z{{\bf Z}}\Ch_{≥0}(\Z)$ refers to the derived ∞-category of chain complexes, i.e., with quasi-isomorphisms inverted up to a homotopy.
Recall that the ri …
- 37.8k
6
votes
Accepted
Is there a Čech-like way of computing $H^\bullet(X,M^\bullet)$ or even $\mathsf{R}f_* M^\bul...
Is there a Cech-like way of describing the (hyper)cohomology H∙(X,M∙) or, even better, the complex Rf∗M∙ for some map f?
Yes, the Verdier hypercovering theorem allows one to compute sheaf cohomology …
- 37.8k
4
votes
Accepted
Are there Alexander-Whitney and shuffle maps for Dold-Kan for abelian categories?
§1.2.3 in Lurie's Higher Algebra establishes the Dold—Kan correspondence for idempotent complete additive categories
and constructs the Alexander—Whitney maps in this generality.
- 37.8k
5
votes
Accepted
Derived functors out of an unbounded derived $\infty$-category
An account of derived functors between ∞-categories equipped with weak equivalences and fibrations can be found in Section 7.5 of Cisinski's Higher Categories and Homotopical Algebra. This setting is …
- 37.8k
10
votes
Applications of the Dold-Kan correspondence
In addition to the construction of (generalized) Eilenberg–MacLane spaces mentioned in the comments, there are many other examples.
For example, the Dold–Kan correspondence allows us an easy perspecti …
- 37.8k
14
votes
Examples of topoi that are not ordinary spaces
The bicategory of Grothendieck toposes is equivalent to the bicategory
of localic groupoids, with appropriately defined 1-morphisms and 2-morphisms.
"Ordinary spaces" are topological spaces, or, perh …
- 37.8k
2
votes
Hochschild/cyclic homology of von Neumann algebras: useless?
I am surprised that nobody mentioned the book by Sinclair and Smith “Hochschild Cohomology of von Neumann algebras” so far.
If I remember it correctly, they claim that nobody knows whether there is a …
- 37.8k
6
votes
Accepted
Where can I find a definition of $\underline{H}^p(X, \mathscr{F})$?
As indicated in the comments, the notation $\def\HH{\underline{\rm H}}\def\H{{\rm H}}\HH^p(X,F)$ is defined (for example) by Milne in Étale Cohomology as the $p$th right derived functor of the inclusi …
- 37.8k
11
votes
1
answer
495
views
Is there a practical criterion to determine whether the limit of a diagram of real chain com...
Consider a diagram D: I→ChR of real connective chain complexes.
In the example I have in mind all chain complexes are concentrated in some fixed degree n.
There is a canonical map lim D → holim D fro …
- 37.8k
5
votes
Accepted
Reference for homotopy colimit = total complex
See Problem 4.23 and Problem 4.24 (with proofs) of Ulrich Bunke's Differential cohomology.
The underlying abstract machinery for computing homotopy (co)limits
via homotopy (co)ends is presented by
Se …
- 37.8k