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$\DeclareMathOperator\Sq{Sq}$The Steenrod squares $\Sq^i: H^n({-};\mathbb{F}_2) \to H^{n+i}({-};\mathbb{F}_2)$ are fundamental cohomological operations. By the Yoneda lemma, they induce a map between …

And again can you connect the modern treatment of nonabelian cohomology with this classical picture? …

Summary Someone claims $\mathbb{R}$ can be constructed as the following intriguing quotient, which is related to Gromov's bounded cohomology. I want to find out if it is true. … Related Category-theoretic description of the real numbers (Mathematics Stack Exchange) Gromov's bounded cohomology, see Ivanov - Notes on the bounded cohomology theory and the 9th page of A'Campo's paper …

Let $F$ be a finite field, $Sq^i$ be the $i$-th Steenrod operation $$ H^*(-;F) \to H^{*+i}(-;F).$$ By Yoneda lemma, such operation is a map $\phi_i: B^{*}F \to B^{*+i} F$, where $B$ denotes the delo …

Given a simply connected topological space $X$, it is well known that its free loop space $LX$ has cohomology being the Hochschild homology of the singular cochains [1]: $$HH_\bullet(S^\star X) \simeq …

In the nPOV, cohomology is realized as the connected component of the derived hom space [1]. … References [1] nLab: cohomology [2] nLab: derived functor …

There's definitely a deep relation among obstruction theory, extension problem, cohomology theory, deformation theory, .. etc Some relating urls could be helpful: Deligne's letter to Miller (mathoverflow.net … (mathieu.anel.free.fr/mat/doc/…) John Baez's lecture note "n-category and cohomology" (http://www.math.uchicago.edu/~may/IMA/BaezShulman.pdf) In particular, roughly speaking: (1) Deligne pointed out …

Given a Lie algebra $\frak{g}$, a linear representation $V$, and a 3-cohomology class $\alpha \in H^3(\frak{g}$$, M)$ we can construct a Lie 2-algebra. … Namely, do $(n+1)$-th cohomology classifies Lie n-algebras, if any? If so, how does do we connect this back to the classical extension theory? …

As $B$ varies, I believe the question is equivalent to Question': To what extent does the cohomology ring $H^\star(BG;\mathbb{Z})$ fail to characterize $BG$ up to homotopic equivalence. … It is well-known that taking cohomology does forget much information. And for good enough spaces still, one needs to consider the cochain complex as an $E_\infty$ algebra. …

In this case, the monoidal structures and the braided structures (up to suitable equivalence) can be classified by some cohomology classes. …

For a noncommutative semisimple ring $R$, its structure and its category of representations can be largely understood using Artin-Wedderburn theorem. Such structure theory is useful, for example, in t …

From this, one derives a cohomology theory of this algebraic theory. … This subsumes group cohomology, Lie algebra cohomology, Hochschild cohomology, and Harrison's cohomology for commutative algebras [2, chapter 6+7]. …