nLab
cochain on a simplicial set

Idea

The collection $[S^{•},R]$ of $R$-valued functions on a simplicial set $S^{•}$ is a commutative cosimplicial algebra. Under the monoidal Dold–Kan correspondence it maps to its Moore cochain complex $C^{•}([S^{•},R])$ 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.

Definition

Let $R$ be commutative ring.

For $S$ a set, write

$$[S,R]=R^S$$[S,R] = R^S

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 $S=(S_{•}):{\Delta }^{\mathrm{op}}\to \mathrm{Set}$ a simplicial set write $[S_{•},R]$ for the cosimplicial algebra obtained by taking $R$-valued functions in each degree. This is naturally

• a cosimplicial group

• and even a cosimplicial $R$-module

• and even a cosimplicial algebra over $R$ .

Equivalently, if we write $R[S_{•}]$ for the simplicial $R$-module which is in degree $n$ the free $R$-module on the set $S_n$, we have a canonical isomorphism

$$[S_{•},R]\simeq {\mathrm{Hom}}_{R\mathrm{Mod}}(R[S_{•}],R).$$[S_\bullet,R] \simeq Hom_{R Mod}(R[S_\bullet], R) \,.

This latter point of view is often preferred in the literature when $R[S_{•}]$ is regarded as the collection of chains on $S$ and $[S_{•},R]$ as that of cochains .

More precisely, we should speak of chains and cochains after applying the Moore complex functor. Write

$$C^{•}(S,R):=C^{•}([S_{buller},R])$$C^\bullet(S,R) := C^\bullet([S_\buller,R])

for the Moore cochain complex obtained from the cosimplicial group $[S_{•},R]$. This is the cochain complex of the simplicial set $S$. Using the cup product, this is even a dg-algebra.

Properties

Proposition

The functor

$$[-,R]:\mathrm{SSet}\to [{\Delta }^{op},R\mathrm{Mod}]$$[-,R] : SSet \to [\Delta^\op,R Mod]

is a symmetric lax monoidal functor.

Proof

For instance Prop 3.8 in (May03) .

…

Homotopy commutativity

The dg-algebra of cochains $C^{•}(S,R)$ is not, in general, (graded) commutative. But it is homotopy commutative in that it is an algebra over an operad for an E-∞ operad.

Theorem

The cochain functor

$$C^{•}[-,R]:\mathrm{SSet}\to \mathrm{dgAlg}$$C^\bullet[-,R] : SSet \to dgAlg

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 $C^{•}(S,R)\otimes C^{•}(S,R)\to C^{•}(S,R)$ is the usual cup product.

Proof

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) .

Examples

• For $X$ a topological space and ${\Delta }_{\mathrm{Top}}:\Delta \to \mathrm{Top}$ the canonical topological simplices, the simplicial set $X^{{\Delta }_{\mathrm{Top}}^{•}}$ is the singular simplicial complex of $X$. It cochain dg-algebra is the one that computes the singular cohomology of $X$.

References

An explicit description of the cochains that express the homotopy symmetry of the cup product is given from page 30 on of the old

• Norman Steenrod, Cohomology operations and obstructions to extending continuous functions , Colloquium lectures (1957)

The modern operad-theoretic statement that for $S\in$ SSet a simplicial set, the cochain complex $C^{•}([S,R])$ is an E-∞ algebra apparently goes back to

• V. Hinich and V. Schechtman, On homotopy limits of homotopy algebras, in K-theory, arithmetic and geometry, Lecture notes Vol. 1289, Berlin 1987 pp. 240–264

A particularly clear exposition is in

• Michael Mandell, Cochain multiplication, Amer. J. Math. 124 (2002)

This in turn is nicely reviewed and spelled out in section 3 of

• Peter May, Operads and sheaf cohomology (2003) (unpublished private notes – but maybe we get permission to upload them here?)

These describe actions of the Eilenberg–Zilber operad on $C^{•}([S^{•},R])$.

An action of instead the Barratt-Eccles operad is described in

• Clemens Berger, Benoit Fresse Combinatorial operad actions on cochains (arXiv:0109158)