nLab cochain on a simplicial set


This entry may need to be merged with simplicial cochain.




Special and general types

Special notions


Extra structure



Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

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see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




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

For SS a set, write

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

for the RR-valued functions on SS: the set of maps from SS to RR (using either internal hom notation or exponential object notation).

This is in particular naturally

  • a group (using the addition in RR);

  • and even an RR-module

  • and even an RR-algebra.

  • and even a commutative RR algebra (since RR is assumed to be commutative ring).

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

Equivalently, if we write R[S ]R [S_\bullet] for the simplicial RR-module which is in degree nn the free RR-module on the set S nS_n, we have a canonical isomorphism

[S ,R]Hom RMod(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 ]R[S_\bullet] is regarded as the collection of chains on SS and [S ,R][S_\bullet,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 ,R]) C^\bullet(S,R) := C^\bullet([S_\bullet,R])

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



The functor

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

is a symmetric lax monoidal functor.


For instance Prop 3.8 in (May03) .

Homotopy commutativity

The dg-algebra of cochains C (S,R)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

C [,R]:SSetdgAlg 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)C (S,R)C (S,R)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 SS \in SSet a simplicial set, the cochain complex C ([S,R])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

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

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

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

Last revised on February 1, 2021 at 02:32:05. See the history of this page for a list of all contributions to it.