symmetric monoidal (∞,1)-category of spectra
In categorical algebra, the bar construction takes an algebra of a monad and systematically “puffs it up”, replacing it with a simplicial object in which all equations in the original algebra are replaced by 1-simplices, all equations between equations (called syzygies) are replaced by 2-simplices, and so on.
More precisely, the bar construction takes an algebra of a monad on a category to an augmented simplicial object in the Eilenberg-Moore category of that monad. The face and degeneracy maps of this augmented simplicial object are given by the structure maps of the monad and its action on the algebra :
If we apply the forgetful functor to , we get an augmented simplicial object in called the bar resolution of :
This is called a resolution of because it is equipped with a simplicial homotopy equivalence to the underlying -object , in a sense clarified by the “acyclic structure” of Definition . Moreover the bar resolution of has a universal property: it is initial among resolutions of , as explained in Theorem .
Here we distinguish between the bar construction and the bar resolution, although the terms are often used interchangeably. It is worth noting that the bar construction is not, in general, simplicially homotopy equivalent to in the category of simplicial objects in . Only after applying the forgetful functor do we obtain a key feature of the bar resolution: the simplicial homotopy equivalence .
(Above we are identifying objects in a category, either or , with simplicial objects in that category that are “discrete”.)
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
The composite
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 .
To explain the sense in which is a 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.
Definition gives an absolute notion of acyclicity, in the sense that if carries an acyclic structure and is any functor, then automatically carries an acyclic structure . (For example, becomes acyclic in a standard model category sense under any functor .)
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
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 , 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 .
The preceding proof does not show that the fit together to form a map of simplicial -algebras (i.e., to respect faces and degeneracies); it merely shows at most one such -algebra resolution map is possible. But once we show that respects faces and degeneracies, the proof of Theorem will be complete.
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 , 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 .
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 .
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.
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 .
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.
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.
There is a monad on whose algebras are left -sets, given by
The terminal -set , namely the singleton set equipped with the trivial action of , is an algebra for this monad .
Now recalling that the bar construction takes any algebra of any monad and turns it into a simplicial object in the category of algebras of that monad, write for the augmented simplicial set obtained by applying this to .
By Theorem , its underlying simplicial set is contractible. This simplicial set is often denoted (and as such known as the universal simplicial -principal bundle) or (see here at simplicial classifying space).
Working through the details, we see that the set of -simplices of is , and all the face maps are given by multiplying successive entries in the -tuple, except for the last face map, which encodes the trivial action of on :
for , while
The quotient is known as the simplicial classifying space of (also denoted ), and its topological realization is a topological classifying space for -principal bundles. Finally the topological realization of the quotient map is the “universal -principal bundle”.
The simplicial set is also the simplicial nerve of the delooping groupoid of .
Let be a commutative associative algebras over some ring . Write for the category of right modules over .
For a right module, also is canonically a right module. This construction extends to a functor
The monoid structure on makes into a monad on : the monad product and unit are given by the product and unit in .
For any right -module , the action makes into an algebra of the monad . The bar construction is then the simplicial -module
Under the Moore complex functor of the Dold-Kan correspondence, this simplicial -module is identified with a chain complex of -modules 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. It got its name because the first authors denoted its elements using a notation involving vertical bars (Ginzburg).
This chain complex provides a free resolution of , which can be used to compute the Tor group
This gives the Hochschild homology of . See there for more details.
See bar and cobar construction.
See (Fresse).
simplicial resolution; this is essentially the same concept but from a slightly different perspective.
The original reference for bar constructions in the generality of monads is
A general discussion of bar construction for monads is at
Textbook accounts can be found at:
Saunders Mac Lane, section IV.5 of Homology
Charles Weibel, section 8.6 of An Introduction to Homological Algebra (1994)
The bar complex of a bimodule is reviewed for instance in
around page 16.
The bar complex for E-infinity algebras is discussed in
The compositional structure of the bar construction of several monads, as well as its interpretation in terms of partial evaluations is studied in
N.B.: including the empty ordinal. ↩
If is a simplicial object, then is what is usually denoted , the object of cells in dimension . Note that is the augmented component. The can be thought of as the number of vertices of a simplex of dimension . We choose the index over the geometric dimension as it is more convenient for our purposes. ↩
Last revised on June 12, 2024 at 16:46:11. See the history of this page for a list of all contributions to it.