nLab cochain on a simplicial set

This entry may need to be merged with simplicial cochain.

Idea

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.

Definition

Let $R$ be commutative ring.

For $S$ a set, write

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

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

• a cosimplicial group

• 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

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

This latter point of view is often preferred in the literature when $R[S_\bullet]$ 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

$$C^\bullet(S,R) := C^\bullet([S_\bullet,R])$$

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.

Properties

Proposition

The functor

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

is a symmetric lax monoidal functor.

Proof

For instance Prop 3.8 in (May03) .

…

Homotopy commutativity

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

Theorem

The cochain functor

$$C^\bullet[-,R] : SSet \to dgAlg$$

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 $C^\bullet(S,R) \otimes C^\bullet(S,R) \to C^\bullet(S,R)$ 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 $\Delta_{Top} : \Delta \to Top$ the canonical topological simplices, the simplicial set $X^{\Delta^\bullet_{Top}}$ is the singular simplicial complex of $X$. It cochain dg-algebra is the one that computes the singular cohomology of $X$.

Related concepts

• chain homology

References

Basics are for instance in Application 1.1.3 of

• Charles Weibel, An Introduction to Homological Algebra

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 $C^\bullet([S,R])$ 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).

  PDF. (unpublished private notes – but maybe we get permission to upload them here?)

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

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

An action of a sequence operad, which is isomorphic to the surjection operad? of Berger and Fresse was constructed by

• James E. McClure, Jeffrey H. Smith, Multivariable cochains operations and little $n$-cubes_ (arXiv:math/0106024)