Algebras and modules
Model category presentations
Geometry on formal duals of algebras
The bar construction takes a monad equipped with an algebra-over-a-monad to the (augmented) simplicial object
This simplicial object is a resolution of .
Regard as a constant simplicial object. The canonical morphism
is a resolution of .
In fact, the bar construction is the universal resolution in the sense of
For modules over an algebra
Let be a commutative associative algebras over some ring . Write for the category of connective chain complexes of modules over .
For a right module, also is canonically a module. This construction extends to a functor
The monoid-structure on makes this a monad in Cat: the monad product and unit are given by the product and unit in .
For a module its right action makes the module an algebra over this monad.
The bar construction is then the simplicial module
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
This gives the Hochschild homology of . See there for more details.
For differential graded (Hopf) algebras
See bar and cobar construction.
A general discussion of bar construction for monads is at
The bar complex of a bimodule is reviewed for instance in
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