group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
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homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
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see also algebraic topology
Introductions
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Basic facts
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The collection $[S^\bullet,R]$ of $R$-valued functions on a simplicial set $S^\bullet$ is a commutative cosimplicial algebra. Under the monoidal Dold–Kan correspondence it maps to its Moore cochain complex $C^\bullet([S^\bullet,R])$ which is a dg-algebra under the cup product: this is the cochain complex of the simplicial set.
Notably, this cochain complex is an E-∞ algebra (an algebra over the E-∞ operad). In cohomology it becomes a graded-commutative algebra.
Let $R$ be commutative ring.
For $S$ a set, write
for the $R$-valued functions on $S$: the set of maps from $S$ to $R$ (using either internal hom notation or exponential object notation).
This is in particular naturally
a group (using the addition in $R$);
and even an $R$-module
and even an $R$-algebra.
and even a commutative $R$ algebra (since $R$ is assumed to be commutative ring).
Similarly, for $S = (S_\bullet) : \Delta^{op} \to Set$ a simplicial set write $[S_\bullet,R]$ for the cosimplicial algebra obtained by taking $R$-valued functions in each degree. This is naturally
and even a cosimplicial $R$-module
and even a cosimplicial algebra over $R$ .
Equivalently, if we write $R [S_\bullet]$ for the simplicial $R$-module which is in degree $n$ the free $R$-module on the set $S_n$, we have a canonical isomorphism
This latter point of view is often preferred in the literature when $R[S_\bullet]$ is regarded as the collection of chains on $S$ and $[S_\bullet,R]$ as that of cochains .
More precisely, we should speak of chains and cochains after applying the Moore complex functor. Write
for the Moore cochain complex obtained from the cosimplicial group $[S_\bullet,R]$. This is the cochain complex of the simplicial set $S$. Using the cup product, this is even a dg-algebra.
For instance Prop 3.8 in (May03) .
…
The dg-algebra of cochains $C^\bullet(S,R)$ is not, in general, (graded) commutative. But it is homotopy commutative in that it is an algebra over an operad for an E-∞ operad.
The cochain functor
naturally factors through algebras over an E-∞ operad, notably the Eilenberg–Zilber operad as well as the Barratt-Eccles operad.
In both these cases the complex of binary operations in these operads has a 0-cycle whose action $C^\bullet(S,R) \otimes C^\bullet(S,R) \to C^\bullet(S,R)$ is the usual cup product.
The statement for the Eilenberg–Zilber operad goes back to HinSch87 . A good review is in (May03) . The statement for the Barrat–Eccles operad is in (BerFre01) .
Basics are for instance in Application 1.1.3 of
An explicit description of the cochains that express the homotopy symmetry of the cup product is given from page 30 on of the old
The modern operad-theoretic statement that for $S \in$ SSet a simplicial set, the cochain complex $C^\bullet([S,R])$ is an E-∞ algebra apparently goes back to
A particularly clear exposition is in
This in turn is nicely reviewed and spelled out in section 3 of
These describe actions of the Eilenberg–Zilber operad on $C^\bullet([S^\bullet,R])$.
An action of instead the Barratt-Eccles operad is described in