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bar and cobar construction

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Bar and cobar constructions

  • There is a brief entry at bar construction together with a blog link

  • There is some discussion of the bar-cobar adjointness as it related to twisting cochains, at that entry.

  • Here we will concentrate on the bar-cobar adjointness itself and start exploring the links with other parts of differential algebra.

One of the earliest examples of a pair of adjoint functors studied in algebraic topology was that giving the relationship between the functors for reduced suspension and based loop space. If we take a pointed connected space (X,x 0)(X,x_0), then its reduced suspension ΣX\Sigma X is obtained by taking the cylinder I×XI\times X and identifying the subspace {0,1}×XI×{x 0}\{0,1\}\times X\cup I\times \{x_0\} to a point. (Think of crushing the two ends of the cylinder and the line through the base point to a point.) This can also be thought of as forming S 1XS^1\wedge X the smash product of the circle with XX.

Adjoint to Σ\Sigma is the loop space functor: ΩY\Omega Y is the space of pointed maps from S 1S^1 to YY. This has a monoid structure (up to homotopy) given by concatenation of loops. (Back in S 1S^1, we have a comonoid structure with respect to the pointed coproduct S 1S 1S 1S^1\to S^1\vee S^1 as described at interval object. This in some sense is ‘subdivision as an inverse for composition’.)

(perhaps: Picture to go here?)

Using ordinary (co)homology to study spaces such as CW-complexes, we naturally use the complexes of (cellular) chains on spaces. The structure of chains on the suspension is easy to work out using the obvious cellular structure, but that on the loop space is much harder as ΩX\Omega X is given the compact open topology and only has the homotopy type of a CW-complex, so no nice cellular structure is given us ‘on a plate’. The idea is thus to start with a chain complex model, C *(X)C_*(X), for a CW-complex, XX, (usually the complex of cellular chains on XX), and we try to construct from C *(X)C_*(X) a ‘model’ for the chain complex of the loop space ΩX\Omega X of XX. Adams’ cobar construction was such a method (see below). This was adjoint to a bar construction defined by Eilenberg and MacLane.

Both directions use an abstract algebraic model of concatenation of paths and so their construction is linked to that of free monoids, and through those to monads, operads and related abstract machinery to handle concatenation and its higher categorical analogues in categorical contexts.

The chain complex C *(X)C_*(X) has a rich coalgebraic structure coming from induced by a cellular diagonal approximation on XX so the cobar construction will start with a dg-coalgebra as ‘input’ and as output we will hope for both a coalgebra structure (reflecting the chain coalgebra idea) and an algebra structure (coming from modelling the concatenation of loops). We therefore might hope for, and in fact do get, a differential graded Hopf algebra.

Going the other way, we start with a differential graded algebra and use ‘coconcatenation’ or ‘subdivision’ to get a coalgebra structure. In fact, once again, this is a Hopf algebra.

These topologically motivated constructions can be applied in much greater generality as we will see both here and elsewhere:

The Bar construction

(due originally Eilenberg-MacLane) Remember this goes from ‘algebras’ to Hopf algebras in general.

(1)B:preεCDGApreCDGHAB :pre \varepsilon CDGA \to pre CDGHA

Urs: This is a bit abrupt. Could we just say this in words, once, like “The bar construction, originally due to MacLane, is a map that sends xyz to abc such that rst.”? I would also greatly appreciate some more motivational background and other helpful side remarks. Such as: why does the bar/cobar-construction go between algebras and coalgebras, instead of being an endomorphism of either?

Tim: I have tried to motivate this a bit above. This needs more filling in (and also I think that (slowly) I will go through changing pre-gvs etc to gvs and then talk about positively or negatively graded gvs and so on.)

Definition

Let (A,d,ε)(A,d,\varepsilon) be a commutative, augmented differential \mathbb{Z}-graded algebra, d(A n)A n1d(A_n)\subseteq A_{n-1}, A¯=Kerε\overline{A} = Ker \varepsilon.

The bar construction B(A,d,ε)B(A,d,\varepsilon) is given by

B(A,d,ε)=(T(sA¯),D),B(A,d,\varepsilon) = (T(s\overline{A}), D),

where

d I(sa 1sa n)= i=1 nη(i1)sa 1sa i2sda i1sa n,d_I(sa_1\otimes \ldots\otimes sa_n) = -\sum_{i = 1} ^n\eta(i-1)sa_1\otimes \ldots \otimes sa_{i-2}\otimes sda_{i-1}\otimes\ldots sa_n,

and

d E(sa 1sa n)= i=1 nη(i1)sa 1sa i2sa i1.a isa n,d_E(sa_1\otimes \ldots\otimes sa_n) = -\sum_{i = 1} ^n\eta(i-1)sa_1\otimes \ldots \otimes sa_{i-2}\otimes sa_{i-1}.a_i\otimes \ldots sa_n,

with η(i)=(1) k=1 i|sa k|\eta(i) = (-1)^{\sum_{k=1}^i |sa_k|}.

Note that the image of a 1-connected cdga is a connected commutative Hopf algebra.

Things to note in this construction:

  • It uses the suspension operator on the graded vector spaces. This mirrors the reduced suspension at the cell complex level.

  • It uses a tensor algebra construction. This from one point of view handles the formal concatenation aspect,
    but has also a rich structure of a coalgebraic structure with reduced diagonal, given by

Δ¯(v 1v n)= p=1 n1(v 1v p)(v p+1v n),\bar{\Delta}(v_1\otimes \ldots \otimes v_n) = \sum_{p=1}^{n-1} (v_1\otimes \ldots \otimes v_p)\otimes(v_{p+1}\otimes \ldots \otimes v_n),

(see differential graded coalgebra). This can be interpreted as looking at how a formal concatenation can be ‘subdivided’ into its various parts.

The Cobar construction

(due to J. F. Adams)

We define a functor:

(2)F:preηCoDGCpreCoDGHAF :pre \eta CoDGC \to pre CoDGHA

so essentially from cocommutative differential graded coalgebras to cocommutative differential graded Hopf algebras (with frills attached in the way of coaugmentations, etc).

Let (C,,η)(C,\partial,\eta) be a cocommutative differential \mathbb{Z}-graded coaugmented coalgeba:

(C n)C n1,C¯=C/η(k),Δ¯:C¯C¯C¯.\partial(C_n) \subseteq C_{n-1}, \quad \overline{C} = C/\eta(k), \quad \overline{\Delta} : \overline{C} \to \overline{C}\otimes \overline{C}.

The Cobar construction F(C,,η)F(C,\partial, \eta) is the cocommutative pre-dgha defined by

  • F(C,,η)=(T(s 1C¯),δ)F(C,\partial,\eta) = (T(s^{-1}\overline{C}), \delta), where δ= I+ E\delta = \partial_I + \partial_E.

Here

  • T(s 1C¯)T(s^{-1}\overline{C}) is the cocommutative Hopf algebra generated by s 1C¯s^{-1}\overline{C}, as before(in differential graded coalgebra) C¯\overline{C} is the cokernel of the coaugmentation, η\eta)

  • I(s 1c 1s 1c n)= i=1 nη(i1)s 1c 1s 1c i1s 1c is 1c n,\partial_I (s^{-1}c_1\otimes \ldots\otimes s^{-1}c_n) = -\sum_{i = 1} ^n\eta(i-1)s^{-1}c_1\otimes \ldots\otimes s^{-1}c_{i-1}\otimes s^{-1}\partial c_i\otimes \ldots s^{-1}c_n,

and

  • E(s 1c 1s 1c n)= i=1 nη(i1) μ(1) |c iμ|+1(s 1c 1s 1c iμs 1c iμ s 1c n),\partial_E (s^{-1}c_1\otimes \ldots\otimes s^{-1}c_n) = -\sum_{i = 1} ^n\eta(i-1)\sum_\mu (-1)^{|c'_{i\mu}| +1} (s^{-1}c_1\otimes \ldots\otimes s^{-1}c'_{i\mu}\otimes s^{-1}c^{\prime\prime}_{i\mu}\otimes \ldots\otimes s^{-1}c_n),

    with Δ¯c i= μc iμc iμ \overline{\Delta}c_i = \sum_\mu c'_{i\mu}\otimes c^{\prime\prime}_{i\mu}; η(i)=(1) k=1 i|s 1c k|.\eta(i) = (-1)^{ \sum^i_{k=1}|s^{-1}c_k|}.

The image of a 1-connected cdgc is a connected cocommutative dgha.

If CC is of finite type, #F(C,,η)\#F(C,\partial,\eta) is isomorphic to B#(C,,η)B\#(C,\partial,\eta) as a differential \mathbb{Z}-graded Hopf algebra.

If AA is not (graded) commutative, the differential d Ed_E of B(A,d,ε)B(A,d,\varepsilon) does not respect the shuffle product on T(sA¯)T(s\overline{A}); B(A,d,ε)B(A,d,\varepsilon) thus becomes merely a differential \mathbb{Z}-graded coalgebra. Similarly if CC is not (graded) cocommutative F(C,,η)F(C,\partial,\eta) is merely a differential \mathbb{Z}-graded algebra.

In particular, let

  • εDGA\varepsilon-DGA be the category of augmented differential graded algebras, (A= p0A pA = \oplus_{p\geq 0}A_p).

  • DGC 0DGC_0, the category of connected differential graded coalgebras,

then the Bar and Cobar constructions yield functors

B:εDGADGC 0B: \varepsilon DGA\to DGC_0
F:DGC 0εDGA.F : DGC_0\to \varepsilon DGA.
Proposition

(Husemoller-Moore-Stasheff)

BB is right adjoint to FF.

For any objects (A,d)(A,d) in εDGA\varepsilon-DGA, and (C,)(C,\partial) of DGC 0DGC_0, the natural adjunction morphisms

α^:FB(A,d)(A,d)\hat{\alpha} : FB(A,d) \to (A,d)
β^:(C,)BF(C,)\hat{\beta} : (C,\partial) \to BF(C,\partial)

are weak equivalences / quasi-isomorphisms.

These latter morphisms are defined by

  • α^:T(s 1T(sA¯)¯),δ)(A,d)\hat{\alpha} : T(s^{-1}\overline{T(s\overline{A})}), \delta)\to (A,d) is the zero mapping on s 1T 2(sA¯)s^{-1}T^{\geq 2}(s\overline{A}) and the natural isomorphism s 1sA¯A¯s^{-1}s\overline{A} \stackrel{\simeq}{\to} \overline{A} on s 1sA¯s^{-1}s\overline{A}.

  • β^:(C,)(T(sT(s 1C¯¯),D)\hat{\beta} : (C,\partial) \to (T(\overline{sT(s^{-1}\overline{C}}),D) is the unique lifting of

    Cs 1C¯T(s 1C¯)¯sT(s 1C¯)¯.C\to s^{-1}\overline{C} \to \overline{T(s^{-1}\overline{C})}\to \overline{sT(s^{-1}\overline{C})}.

References

The source used for the above was

D. Tanré, Homotopie rationnelle: Modèles de Chen, Quillen, Sullivan, Lecture Notes in Maths No. 1025, Springer, 1983.

This was augmented with material from

H. J. Baues, Geometry of loop spaces and the cobar construction, Mem. Amer. Math. Soc. 25 (230) (1980) ix+171.

Revised on March 10, 2011 06:38:57 by Tim Porter (95.147.237.143)