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The bar construction takes a monad equipped with an algebra-over-a-monad to the (augmented) simplicial object whose structure maps are given by the structure maps of the monad and its action on its algebra:
This simplicial object can be viewed as a resolution of , in a sense explained below.
Let be a category and let be a monad on . We let denote the category of -algebras, and the forgetful functor which is monadic, with left adjoint .
Recall that the (augmented) simplex category , viz. the category consisting of finite ordinals1 and order-preserving maps, is the “walking monoid”, i.e., is initial among strict monoidal categories equipped with a monoid object. The monoidal product on is ordinal addition . If is the -element ordinal, then the terminal object carries a unique monoid structure and represents the “generic monoid”2.
Similarly is the walking comonoid. Since the comonad on can be regarded as a comonoid in the strict monoidal category of endofunctors (with endofunctor composition as monoidal product), there is a strong monoidal functor (or in fact a unique strict monoidal functor)
that takes the generic monoid to and generally to .
If furthermore is a -algebra, there is an evaluation functor
and we have the following definition:
The bar construction is the simplicial -algebra given by the composite functor
will here be called the bar resolution of .
In the notation of two-sided bar constructions, the bar construction would be written as , and the bar resolution as .
Décalage and resolutions
To explain the sense in which is an acyclic resolution of (the constant simplicial object) , we recall the fundamental décalage construction. Very simply, putting
the décalage functor on simplicial objects (valued in a category ) is the functor
Note that has a comonad structure (inherited from the comonoid structure on in , see also at décalage – comonad structure), and therefore also carries a comonad structure. Notice also that there is a comonad map (where is left adjoint to since is initial in ), induced by the evident natural inclusion in . This in turn induces a comonad map where is the composite (“discretization”):
The notation is chosen because décalage is essentially a kind of path space construction, i.e., in the case it is a simplicial sets analogue of a topological pullback
where is the identity inclusion of the underlying set with the discrete topology. is essentially a sum of spaces of based paths over all possible choices of basepoint , fibered over by taking to . Each space of based paths is contractible and therefore is acyclic.
The following definition names a nonce expression; this author (Todd Trimble) does not know how this is (or might be) referred to in the literature:
An acyclic structure on a simplicial object is a -coalgebra structure .
Here a -coalgebra structure on is the same as a right -coalgebra (or -comodule) structure, given by a simplicial map satisfying evident equations. In more nuts-and-bolts terms, it consists of a series of maps satisfying suitable equations.
The map may be viewed as a homotopy. Again, turning to the topological analogue for intuition, the corresponding is a homotopy (or rather, the composite can be turned into a homotopy ). The coalgebra structure has a retraction given by the counit , so becomes a retract of an acyclic space, hence acyclic itself.
Returning now to the bar resolution : there is a canonical natural isomorphism obtained as the following 2-cell pasting (where abbreviates the top and bottom horizontal composites)
whence there is a homotopy
The map is an acyclic structure, def. 2, i.e., a right -coalgebra structure.
We verify the coassociativity condition for the coaction ; the counit condition is checked along similar lines. The comultiplication of the comonad is , and putting for a right -coaction, the coassociativity of follows from a naturality square
Apply to this coassociativity square to get another coassociativity, this time for the comonad on (with comultiplication denoted ) and coaction . Thus there is an equalizing diagram
Because is a strong monoidal functor (see the left isomorphism in (1)), the squares in
commute serially, with the triangle commuting by definition of . This completes the verification.
By Remark 1, it follows that , obtained by applying evaluation at a -algebra , carries an acyclic structure as well. In this sense we may say that (which has as its augmented component in dimension ) is an acyclic resolution of the constant simplicial -algebra at that carries a -algebra structure.
We now state and prove a universal property of the bar construction .
Let be a monad on a category . A -algebra resolution is a simplicial object together with an acyclic structure on . A morphism between -algebra resolutions is a natural transformation such that is a -coalgebra map.
Let be the category of -algebra resolutions. There is a forgetful functor
that takes an algebra resolution to its augmentation component .
The functor is represented by ; i.e., is left adjoint to .
The proof is distributed over two lemmas.
Given a -algebra resolution and a -algebra map , there is at most one -algebra resolution map such that .
The -coalgebra structure is defined on components by . Thus in order that be a -coalgebra map, we must have that the diagram
commutes. Here determines a unique -algebra map such that
since is left adjoint to . Thus, starting with as given, each algebra map uniquely determines its successor .
Given a -algebra resolution and an algebra map , there is at least one -algebra resolution map with .
It is enough to produce such a map in the case , since the case for general is then given by a composite
We will do something slightly more general. For any category , the endofunctor category has a comonoid object , so that there is an induced strong monoidal functor
which, upon evaluating at an object of , gives a functor
with , so that is a double simplicial object. Taking and taking to be a -algebra resolution with acyclic structure , we will produce a (double) simplicial map
where is defined recursively as in the proof of Lemma 1, by setting and taking the unique simplicial -algebra map such that
commutes for all . Once we verify the claim that respects faces and degeneracies, the same will be true for , whence the proof will be complete by Remark 2.
The claim is proved by induction on . Let be the counit and be the comultiplication. We have face maps
for to , under the special convention that denotes the action . We also have degeneracy maps
for to . We proceed as follows.
To check preservation of face maps, we treat separately the cases where and .
- For , we must check commutativity of the square in
Since all the maps are algebra maps and exhibits as the free algebra on , it suffices to check commutativity around the perimeter. (N.B.: the triangle commutes, even in the case where which we need to start the induction.) By definition of , commutativity of the perimeter boils down to commutativity of
where the triangle commutes by one of the acyclic structure equations.
- For , the commutativity of the right square in
is again by appeal to a freeness argument where we just need to check commutativity of the perimeter, noting commutativity of the left square by naturality and that of the bottom quadrilateral by the recursive definition of . But the perimeter commutes by examining the diagram
(where the middle vertical arrow is ) using the inductive hypothesis in the bottom left parallelogram.
To check preservation of degeneracy maps, we treat separately the cases and .
- For , the commutativity of the top right square in
is by appeal to a freeness argument where we just need to check commutativity of the perimeter (the special case being used to start the induction). But this boils down to commutativity of the diagram
where the bottom right quadrilateral commutes by one of the acyclic structure equations.
- For , the commutativity of the top right square in
is once again by appeal to a freeness argument where we just need to check commutativity of the perimeter. Here it boils down to commutativity of
where the middle vertical arrow is .
This completes the proof.
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