nLab
bar construction

Idea

The bar construction takes a monad $(T,\mu ,ϵ)$ equipped with an algebra-over-a-monad $(A,\rho )$ to the (augmented) simplicial object

$$\mathrm{B}(T,A):=\left(\cdots \overrightarrow{\overrightarrow{\to }}TTA\stackrel{\stackrel{\mu \cdot {\mathrm{Id}}_A}{\to }}{\stackrel{T\cdot \rho }{\to }}TA\stackrel{\rho }{\to }A\right).$$\mathrm{B}(T,A) := \left( \cdots \stackrel{\to}{\stackrel{\to}{\to}} T T A \stackrel{\stackrel{\mu \cdot Id_A}{\to}}{\stackrel{T \cdot \rho}{\to}} T A \stackrel{\rho}{\to} A \right) \,.

This simplicial object is a resolution of $A$.

Properties

Proposition

Regard $A$ as a constant simplicial object. The canonical morphism

$$\mathrm{B}(T,A)\to A$$\mathrm{B}(T,A) \to A

is a resolution of $A$.

In fact, the bar construction is the universal resolution in the sense of

(…)

Special cases

For modules over an algebra

Let $A$ be a commutative associative algebras over some ring $k$. Write $A\mathrm{Mod}$ for the category of connective chain complexes of modules over $A$.

For $N$ a right module, also $N{\otimes }_{k}A$ is canonically a module. This construction extends to a functor

$$A{\otimes }_k(-):A\mathrm{Mod}\to A\mathrm{Mod}.$$A \otimes_k (-) : A Mod \to A Mod \,.

The monoid-structure on $A$ makes this a monad in Cat: the monad product and unit are given by the product and unit in $A$.

For $N$ a module its right action $\rho :N\otimes A\to N$ makes the module an algebra over this monad.

The bar construction $\mathrm{B}(A,N)$ is then the simplicial module

$$\cdots \overrightarrow{\overrightarrow{\to }}N{\otimes }_{k}A{\otimes }_{k}A\stackrel{\stackrel{\mathrm{Id}\otimes \mu }{\to }}{\underset{\rho \otimes \mathrm{Id}}{\to }}N{\otimes }_{k}A.$$\cdots \stackrel{\to}{\stackrel{\to}{\to}} N \otimes_k A \otimes_k A \stackrel{\overset{Id \otimes \mu}{\to}}{\underset{\rho \otimes Id}{\to}} N \otimes_k A \,.

Under the Moore complex functor of the Dold-Kan correspondence this is identified with a chain complex whose differential is given by the alternating sums of the face maps indicated above.

This chain complex is what originally was called the bar complex in homological algebra. Because the first authors denoted its elements using a notation involving vertical bars (Ginzburg)!!

This chain complex provides a resolution that computes the Tor

$$\mathrm{Tor}(N,A\times A).$$Tor(N, A \times A) \,.

This gives the Hochschild homology of $A$. See there for more details.

For differential graded (Hopf) algebras

See bar and cobar construction.

For $E_{\mathrm{\infty }}$-algebras

See (Fresse).

References

A general discussion of bar construction for monads is at

• Todd Trimble, On the Bar Construction (blog)

The bar complex of a bimodule is reviewed for instance in

• Victor Ginzburg, Lectures on noncommutative geometry (arXiv:math/0506603)

around page 16.

The bar complex for E-infinity algebras is discussed in

• Benoit Fresse, The bar complex of an E-infinity algebra Advances in Mathematics Volume 223, Issue 6, 1 April 2010, Pages 2049-2096