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Results tagged with cohomology
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user 22810
A branch of algebraic topology concerning the study of cocycles and coboundaries. It is in some sense a dual theory to homology theory. This tag can be further specialized by using it in conjunction with the tags group-cohomology, etale-cohomology, sheaf-cohomology, galois-cohomology, lie-algebra-cohomology, motivic-cohomology, equivariant-cohomology, ...
3
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1
answer
149
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A "non-abelian excision" statement for mapping out of a space
Let $U \subset A \subset X$ be spaces (in the sense of homotopy theory).
For every pointed space $Y$ restriction maps induce the following canonical map between mapping spaces:
$$fiber(Map(X,Y) \to …
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5
votes
0
answers
219
views
Is there a systematic way to "bound" the $d_n$'s of ASS's by "pairing" them with elements in...
operation $Sq^{2^{j}}$ acts non-trivialy on the cohomology of some $2$-cell complex. … Question 1: Is there a systematic connection between actions of $s$-order cohomology operations on the cohomology of $2$-cell complexes and persistence of elements in the $s$-line of the ASS? …
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4
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1
answer
875
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When does derived pullback commute with infinite products?
Let $f:X \to Y$ be a morphism of reasonable schemes (qcqs). Let $f^*: D(Y) \to D(X)$ be the pullback defined on the derived unbounded categories of quasi-coherent sheaves.
Question: When does $f^ …
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28
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4
answers
3k
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Yoga of six functors for group representations?
(-)$ is a derived version of $(-)^G$ (invariants) = group cohomology.
$\pi_*(-)$ is a derived version of $(-)_G$ (coinvariants) = group homology .
Generally if $\pi: BH \to BG$ then:
$\pi_! …
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2
votes
2
answers
216
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Continuous map with homeomorphic fibers whose associated $H^{k}_c$ sheaf is not a local system?
Let $ f: X \to Y$ be a continuous map between connected manifolds s.t. for all $y \in Y$ the fiber $f^{-1}(y)$ is homeomorphic to some fixed connected manifold $Z$.
Let $k$ be a ring and for every $ …
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11
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0
answers
551
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The intrinsic meaning of abelian sheaf cohomology of a category
A priori we don't know that $\mathsf{AbC}$ is abelian (we expect it not to be in general) so the composition isn't something we can derive to get cohomology. … In order to define cohomology (so that in particular we get cohomology for abelian group objects) it is reasonable to look for reflective subtoposes of $P$ containing $C$ (image of the yoneda). …
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0
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1
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If $J$-coverings can be glued $I$-locally is $J$-locality an $I$-local property? (Reducing d...
Let $(C,J)$ be a category with a grothendieck topology. For every object $X \in C$ there's (I hope) a little site which is the full subcategory of the slice category $C_{/X}$ whose objects are the mor …
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6
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Categorical formalism for higher non-abelian group cohomology / obstruction theory for gerbes?
So the question is now:
Main question: Is there any hope for a categorical formalism for higher non-abelian group cohomology which would include an obstruction theory for non-abelian gerbes? …
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14
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2
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"Correct" definition of stratified spaces and reference for constructible sheaves?
It seems that the theory of constructible sheaves (in particular anything that goes into proving that they form an abelian category) requires some technical statements about existence of certain strat …
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93
votes
3
answers
10k
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What is homology anyway?
Take this local system $L:X \to \mathrm{Sp}$ and define $L$-cohomology of X to be $\operatorname{Lim} L$ (this coincides with the sheaf cohomology definition) and $L$-homology to be $\operatorname{Colim … Homology $\sim$ dual to Cohomology
This is the most cheeky definition. …
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