Special and general types
Paths and cylinders
The collection of -valued functions on a simplicial set is a commutative cosimplicial algebra. Under the monoidal Dold–Kan correspondence it maps to its Moore cochain complex 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 be commutative ring.
For a set, write
for the -valued functions on : the set of maps from to (using either internal hom notation or exponential object notation).
This is in particular naturally
Similarly, for a simplicial set write for the cosimplicial algebra obtained by taking -valued functions in each degree. This is naturally
Equivalently, if we write for the simplicial -module which is in degree the free -module on the set , we have a canonical isomorphism
This latter point of view is often preferred in the literature when is regarded as the collection of chains on and 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 . This is the cochain complex of the simplicial set . Using the cup product, this is even a dg-algebra.
For instance Prop 3.8 in (May03) .
The dg-algebra of cochains 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 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 SSet a simplicial set, the cochain complex 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
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 .
An action of instead the Barratt-Eccles operad is described in