nLab bar construction




In categorical algebra, the bar construction takes an algebra AA of a monad and systematically “puffs it up”, replacing it with a simplicial object Bar T(A)Bar_T(A) 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 (A,ρ)(A, \rho) of a monad (T,μ,ϵ)(T, \mu, \epsilon) on a category CC to an augmented simplicial object Bar T(A)Bar_T(A) in the Eilenberg-Moore category C TC^T 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 AA:

Bar T(A)(TTAATρAμId ATAρA). Bar_T(A) \coloneqq \left( \cdots \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} T T A \stackrel{\stackrel{\mu \cdot Id_A}{\longrightarrow}}{\stackrel{\phantom{A} T \cdot \rho \phantom{A}}{\longrightarrow}} T A \stackrel{\rho}{\longrightarrow} A \right) \,.

If we apply the forgetful functor U:C TCU : C^T \to C to Bar T(A)Bar_T(A), we get an augmented simplicial object in CC called the bar resolution of AA:

UBar T(A)=(UTTAAU(Tρ)AU(μId A)UTAUρUA). U Bar_T(A) = \left( \cdots \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} U T T A \stackrel{\stackrel{U (\mu \cdot Id_A)}{\longrightarrow}}{\stackrel{\phantom{A} U(T \cdot \rho) \phantom{A}}{\longrightarrow}} U T A \stackrel{U \rho}{\longrightarrow} U A \right) \,.

This is called a resolution of AA because it is equipped with a simplicial homotopy equivalence to the underlying CC-object UACU A \in C, in a sense clarified by the “acyclic structure” of Definition . Moreover the bar resolution of AA has a universal property: it is initial among resolutions of AA, 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 Bar T(A)Bar_T(A) is not, in general, simplicially homotopy equivalent to AA in the category of simplicial objects in C TC^T. Only after applying the forgetful functor U:C TCU \colon C^T \to C do we obtain a key feature of the bar resolution: the simplicial homotopy equivalence UBar T(A)UAU Bar_T(A) \simeq U A.

(Above we are identifying objects in a category, either AC TA \in C^T or UACU A \in C, with simplicial objects in that category that are “discrete”.)


Let E\mathbf{E} be a category and let (T,m:TTT,u:1 ET)(T, m: T T \to T, u: 1_{\mathbf{E}} \to T) be a monad on E\mathbf{E}. We let E T\mathbf{E}^T denote the category of TT-algebras, and U:E TEU: \mathbf{E}^T \to \mathbf{E} the forgetful functor which is monadic, with left adjoint FF.

Recall that the augmented simplex category Δ a\Delta_a, 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 Δ a\Delta_a is ordinal addition [m]+[n]=[m+n][m]+[n] = [m+n]. If [n][n] is the nn-element ordinal, then the terminal object [1][1] carries a unique monoid structure and represents the “generic monoid”2.

Similarly Δ a op\Delta_a^{op} is the walking comonoid. Since the comonad FUF U on E T\mathbf{E}^T can be regarded as a comonoid in the strict monoidal category of endofunctors [E T,E T][\mathbf{E}^T, \mathbf{E}^T] (with endofunctor composition as monoidal product), there is a strong monoidal functor (or in fact a unique strict monoidal functor)

Δ a opBar T[E T,E T]\Delta_a^{op} \stackrel{Bar_T}{\to} [\mathbf{E}^T, \mathbf{E}^T]

that takes the generic monoid [1][1] to FUF U and generally [n][n] to (FU) n(F U)^{\circ n}.

If furthermore AA is a TT-algebra, there is an evaluation functor

[E T,E T]eval AE T[\mathbf{E}^T, \mathbf{E}^T] \stackrel{eval_A}{\to} \mathbf{E}^T

and we have the following definition:


The bar construction Bar T(A)Bar_T(A) is the simplicial TT-algebra given by the composite functor

Δ a opBar T[E T,E T]eval AE T.\Delta_a^{op} \stackrel{Bar_T}{\to} [\mathbf{E}^T, \mathbf{E}^T] \stackrel{eval_A}{\to} \mathbf{E}^T.

The composite

Δ a opBar T(A)E TUE\Delta_a^{op} \stackrel{Bar_T(A)}{\to} \mathbf{E}^T \stackrel{U}{\to} \mathbf{E}

will here be called the bar resolution of AA.

In the notation of two-sided bar constructions, the bar construction would be written as Bar T(A)=B(F,T,A)Bar_T(A) = B(F, T, A), and the bar resolution as B(T,T,A)B(T, T, A).

Décalage and resolutions


To explain the sense in which UBar T(A)U Bar_T(A) is a resolution of (the constant simplicial object) AA, we recall the fundamental décalage construction. Very simply, putting

D=[1]+():Δ a opΔ a opD = [1] + (-): \Delta_a^{op} \to \Delta_a^{op}

the décalage functor on simplicial objects C Δ a opC^{\Delta_a^{op}} (valued in a category C\mathbf{C}) is the functor

P:C Δ a opC DC Δ a op.P: \mathbf{C}^{\Delta_a^{op}} \stackrel{\mathbf{C}^D}{\to} \mathbf{C}^{\Delta_a^{op}}.

Note that DD has a comonad structure (inherited from the comonoid structure on [1][1] in Δ a op\Delta_a^{op}, see also at décalage – comonad structure), and therefore PP also carries a comonad structure. Notice also that there is a comonad map D[1]!D \to [1]\circ ! (where [1]:1Δ a op[1]: 1 \to \Delta_a^{op} is left adjoint to !:Δ a op1!: \Delta_a^{op} \to 1 since [1][1] is initial in Δ a op\Delta_a^{op}), induced by the evident natural inclusion [1]+i:[1]+[0][1]+[m][1]+i: [1]+[0] \to [1]+[m] in Δ a\Delta_a. This in turn induces a comonad map PX|X|P X \to {|X|} where ||{|-|} is the composite (“discretization”):

C Δ a opev [1]CdiagC Δ a op.C^{\Delta_a^{op}} \stackrel{ev_{[1]}}{\to} C \stackrel{diag}{\to} C^{\Delta_a^{op}}.

The notation PP is chosen because décalage is essentially a kind of path space construction, i.e., in the case C=Set\mathbf{C} = Set it is a simplicial sets analogue of a topological pullback

PX X I ev 1X ev 0 |X| id X\array{ P X & \to & X^I & \stackrel{ev_1}{\to} X \\ \downarrow & & \downarrow_\mathrlap{ev_0} & \\ {|X|} & \underset{id}{\to} & X }

where id:|X|Xid: {|X|} \to X is the identity inclusion of the underlying set with the discrete topology. PXP X is essentially a sum of spaces of based paths (α:(I,0)(X,x 0)(\alpha: (I, 0) \to (X, x_0) over all possible choices of basepoint x 0x_0, fibered over XX by taking α\alpha to α(1)\alpha(1). Each space of based paths is contractible and therefore PXP X 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 X:Δ a opCX: \Delta_a^{op} \to C is a PP-coalgebra structure XPXX \to P X.

Here a PP-coalgebra structure on XX is the same as a right DD-coalgebra (or DD-comodule) structure, given by a simplicial map h:XXDh: X \to X \circ D satisfying evident equations. In more nuts-and-bolts terms, it consists of a series of maps h n:X([n])X([n+1])h_n: X([n]) \to X([n+1]) satisfying suitable equations.

The map h:XXDh: X \to X D may be viewed as a homotopy. Again, turning to the topological analogue for intuition, the corresponding h:XPXh: X \to P X is a homotopy (or rather, the composite XPXX IX \to P X \to X^I can be turned into a homotopy I×XXI \times X \to X). The coalgebra structure h:XPXh: X \to P X has a retraction given by the counit ε:PXX\varepsilon: P X \to X, so XX becomes a retract of an acyclic space, hence acyclic itself.


Definition gives an absolute notion of acyclicity, in the sense that if X:Δ a opCX: \Delta_a^{op} \to \mathbf{C} carries an acyclic structure h:XXDh: X \to X D and G:CCG: \mathbf{C} \to \mathbf{C}' is any functor, then GXG X automatically carries an acyclic structure Gh:GXGXDG h: G X \to G X D. (For example, GXG X becomes acyclic in a standard model category sense under any functor G:CSetG: \mathbf{C} \to Set.)


Returning now to the bar resolution UBar T(A)U Bar_T(A): there is a canonical natural isomorphism TUBar TUBar TDT \circ U Bar_T \cong U Bar_T \circ D obtained as the following 2-cell pasting (where UBar TU Bar_T abbreviates the top and bottom horizontal composites)

(1)Δ a op Bar T [E T,E T] [id,U] [E T,E] D=[1]+() [id,FU] [id,UF]=[id,T] Δ a op Bar T [E T,E T] [id,U] [E T,E],\array{ \Delta_a^{op} & \stackrel{Bar_T}{\to} & [\mathbf{E}^T, \mathbf{E}^T] & \stackrel{[id, U]}{\to} & [\mathbf{E}^T, \mathbf{E}] \\ _\mathllap{D = [1] + (-)} \downarrow & \cong & _\mathllap{[id, F U]} \downarrow & \cong & \downarrow_\mathrlap{[id, U F] = [id, T]} \\ \Delta_a^{op} & \underset{Bar_T}{\to} & [\mathbf{E}^T, \mathbf{E}^T] & \underset{[id, U]}{\to} & [\mathbf{E}^T, \mathbf{E}], }

whence there is a homotopy

h=(UBar TuUBar TTUBar TUBar TD).h = (U Bar_T \stackrel{u U Bar_T}{\to} T U Bar_T \cong U Bar_T D).

The map hh is an acyclic structure, def. , i.e., a right DD-coalgebra structure.


We verify the coassociativity condition for the coaction h:UBar TUBar TDh: U Bar_T \to U Bar_T D; the counit condition is checked along similar lines. The comultiplication of the comonad FUF U is δFuU\delta \coloneqq F u U, and putting η=uU:UUFU\eta = u U: U \to U F U for a right FUF U-coaction, the coassociativity of η\eta follows from a naturality square

U η UFU η Uδ UFU ηFU UFUFU.\array{ U & \stackrel{\eta}{\to} & U F U \\ _\mathllap{\eta} \downarrow & & \downarrow_\mathrlap{U\delta} \\ U F U & \underset{\eta F U}{\to} & U F U F U. }

Apply [id E T,][id_{\mathbf{E}^T}, -] to this coassociativity square to get another coassociativity, this time for the comonad K[id E T,FU]K \coloneqq [id_{\mathbf{E}^T}, F U] on [E T,E T][\mathbf{E}^T, \mathbf{E}^T] (with comultiplication denoted δ K\delta_K) and coaction H[id,η]:[id,U][id,U]KH \coloneqq [id, \eta]: [id, U] \to [id, U] \circ K. Thus there is an equalizing diagram

[id,U]H[id,U]KHK[id,U]δ K[id,U]KK.[id, U] \stackrel{H}{\to} [id, U]K \stackrel{\overset{[id, U]\delta_K}{\to}}{\underset{H K}{\to}} [id, U]K K.

Because Bar T:Δ a op[E T,E T]Bar_T: \Delta_a^{op} \to [\mathbf{E}^T, \mathbf{E}^T] is a strong monoidal functor (see the left isomorphism in (1)), the squares in

[id,U]Bar T HBar T [id,U]KBar T HKBar T[id,U]δ KBar T [id,U]KKBar T h [id,U]Bar TD hD[id,U]Bar Tδ D [id,U]Bar TDD\array{ [id, U] Bar_T & \stackrel{H Bar_T}{\to} & [id, U]K Bar_T & \stackrel{\overset{[id, U]\delta_K Bar_T}{\to}}{\underset{H K Bar_T}{\to}} & [id, U]K K Bar_T \\ & _\mathllap{h}{\searrow} & \downarrow_\mathrlap{\cong} & & \downarrow_\mathrlap{\cong} \\ & & [id, U] Bar_T D & \stackrel{\overset{[id, U]Bar_T \delta_D}{\to}}{\underset{h D}{\to}} & [id, U]Bar_T D D }

commute serially, with the triangle commuting by definition of hh. This completes the verification.

By Remark , it follows that UBar T(A)U Bar_T(A), obtained by applying evaluation at a TT-algebra AA, carries an acyclic structure as well. In this sense we may say that UBar T(A)U Bar_T(A) (which has AA as its augmented component in dimension 1-1) is an acyclic resolution of the constant simplicial TT-algebra at AA that carries a TT-algebra structure.


Universal property

We now state and prove a universal property of the bar construction Bar T(A)Bar_T(A).


Let (T,m:TTT,u:1T)(T, m: T T \to T, u: 1 \to T) be a monad on a category E\mathbf{E}. A TT-algebra resolution is a simplicial object Y:Δ a opE TY: \Delta_a^{op} \to \mathbf{E}^T together with an acyclic structure on UY:Δ a opEU Y: \Delta_a^{op} \to \mathbf{E}. A morphism between TT-algebra resolutions is a natural transformation ϕ:YY\phi: Y \to Y' such that Uϕ:UYUYU\phi: U Y \to U Y' is a PP-coalgebra map.

Let AlgRes TAlgRes_T be the category of TT-algebra resolutions. There is a forgetful functor

G:AlgRes TE TG: AlgRes_T \to \mathbf{E}^T

that takes an algebra resolution YY to its augmentation component Y[0]Y[0].


The functor hom E T(A,G):AlgRes TSet\hom_{\mathbf{E}^T}(A, G-): AlgRes_T \to Set is represented by Bar T(A)Bar_T(A); i.e., Bar T():E TAlgRes TBar_T(-): \mathbf{E}^T \to AlgRes_T is left adjoint to GG.

The proof is distributed over two lemmas.


Given a TT-algebra resolution YY and a TT-algebra map f:AY[0]f: A \to Y[0], there is at most one TT-algebra resolution map ϕ:Bar T(A)Y\phi: Bar_T(A) \to Y such that ϕ[0]=f\phi[0] = f.


The PP-coalgebra structure h:UBar T(A)UBar T(A)Dh: U Bar_T(A) \to U Bar_T(A) \circ D is defined on components UBar T(A)[n]=T nAU Bar_T(A)[n] = T^n A by h[n]=uT n(A):T n(A)T n+1(A)h[n] = u T^n(A): T^n(A) \to T^{n+1}(A). Thus in order that UϕU\phi be a PP-coalgebra map, we must have that the diagram

T nA uT nA UFT n(A) Uϕ[n] Uϕ[n+1] UY[n] h Y[n] UY[n+1]\array{ T^n A & \stackrel{u T^n A}{\to} & U F T^n(A) \\ _\mathllap{U\phi[n]} \downarrow & & \downarrow_\mathrlap{U\phi[n+1]} \\ U Y[n] & \underset{h_Y[n]}{\to} & U Y[n+1] }

commutes. Here ϕ[n]:T n(A)Y[n]\phi[n]: T^n (A) \to Y[n] determines a unique TT-algebra map g:FT n(A)Y[n+1]g: F T^n(A) \to Y[n+1] such that

U(g)uT n(A)=h Y[n]Uϕ[n]U(g) \circ u T^n(A) = h_Y[n] \circ U\phi[n]

since FF is left adjoint to UU. Thus, starting with ϕ[0]=f\phi[0] = f as given, each algebra map ϕ[n]\phi[n] uniquely determines its successor ϕ[n+1]=g\phi[n+1] = g.


The preceding proof does not show that the ϕ[n]\phi[n] fit together to form a map ϕ\phi of simplicial TT-algebras (i.e., to respect faces and degeneracies); it merely shows at most one such TT-algebra resolution map is possible. But once we show that ϕ\phi respects faces and degeneracies, the proof of Theorem will be complete.


Given a TT-algebra resolution YY and an algebra map f:XY[0]f: X \to Y[0], there is at least one TT-algebra resolution map ϕ:Bar T(X)Y\phi: Bar_T(X) \to Y with ϕ[0]=f\phi[0] = f.


It is enough to produce such a map ϕ:Bar T(Y[0])Y\phi: Bar_T(Y[0]) \to Y in the case f=1 Y[0]f = 1_{Y[0]}, since the case for general f:XY[0]f: X \to Y[0] is then given by a composite

Bar T(X)Bar T(f)Bar T(Y[0])ϕY.Bar_T(X) \stackrel{Bar_T(f)}{\to} Bar_T(Y[0]) \stackrel{\phi}{\to} Y.

We will do something slightly more general. For any category C\mathbf{C}, the endofunctor category [C Δ a op,C Δ a op][\mathbf{C}^{\Delta_a^{op}}, \mathbf{C}^{\Delta_a^{op}}] has a comonoid object P=DP = - \circ D, so that there is an induced strong monoidal functor

Δ a opP˜[C Δ a op,C Δ a op]\Delta_a^{op} \stackrel{\tilde{P}}{\to} [\mathbf{C}^{\Delta_a^{op}}, \mathbf{C}^{\Delta_a^{op}}]

which, upon evaluating at an object YY of C Δ a op\mathbf{C}^{\Delta_a^{op}}, gives a functor

B(Y,D,D)eval YP˜:Δ a opC Δ a opB(Y, D, D) \coloneqq eval_Y \circ \tilde{P}: \Delta_a^{op} \to \mathbf{C}^{\Delta_a^{op}}

with B(Y,D,D)[n]=YD nB(Y, D, D)[n] = Y D^n, so that B(Y,D,D)B(Y, D, D) is a double simplicial object. Taking C=E T\mathbf{C} = \mathbf{E}^T and taking YY to be a TT-algebra resolution with acyclic structure h:YYDh: Y \to Y D, we will produce a (double) simplicial map

Φ:B(T,T,Y)B(Y,D,D)\Phi: B(T, T, Y) \to B(Y, D, D)

where Φ[n]:T nYYD n\Phi[n]: T^n Y \to Y D^n is defined recursively as in the proof of Lemma , by setting Φ[0]=1 Y\Phi[0] = 1_Y and taking Φ[n+1]:T n+1YYD n+1\Phi[n+1]: T^{n+1} Y \to Y D^{n+1} the unique simplicial TT-algebra map such that

T nY uT nY T n+1Y Φ[n] Φ[n+1] YD n hD n YD n+1\array{ T^n Y & \stackrel{u T^n Y}{\to} & T^{n+1} Y \\ _\mathllap{\Phi[n]} \downarrow & & \downarrow_\mathrlap{\Phi[n+1]} \\ Y D^n & \underset{h D^n}{\to} & Y D^{n+1} }

commutes for all nn. Once we verify the claim that Φ\Phi respects faces and degeneracies, the same will be true for ϕ[n]=Φ[n][0]:(T nY)[0]=T n(Y[0])(YD n)[0]=Y[n]\phi[n] = \Phi[n][0]: (T^n Y)[0] = T^n(Y[0]) \to (Y D^n)[0] = Y[n], whence the proof will be complete by Remark .

The claim is proved by induction on nn. Let ϵ:D1 Δ a op\epsilon: D \to 1_{\Delta_a^{op}} be the counit and δ:DDD\delta: D \to D D be the comultiplication. We have face maps

T jmT nj1Y:T n+1YT nY,YD jϵD nj:YD n+1YD nT^j m T^{n-j-1} Y: T^{n+1} Y \to T^n Y, \qquad Y D^j \epsilon D^{n-j}: Y D^{n+1} \to Y D^n

for j=0j = 0 to nn, under the special convention that mT 1Y:TYYm T^{-1}Y: T Y \to Y denotes the action α:TYY\alpha: T Y \to Y. We also have degeneracy maps

T juT njY:T nYT n+1Y,YD j1δD nj:YD nYD n+1T^j u T^{n-j} Y: T^n Y \to T^{n+1} Y, \qquad Y D^{j-1} \delta D^{n-j}: Y D^n \to Y D^{n+1}

for j=1j = 1 to nn. We proceed as follows.

  • To check preservation of face maps, we treat separately the cases where j=0j = 0 and j1j \geq 1.

    • For j=0j = 0, we must check commutativity of the square in
      T nY uT nY T n+1Y Φ[n+1] YD n+1 id mT n1Y YϵD n T nY Φ[n] YD n.\array{ T^n Y & \stackrel{u T^n Y}{\to} & T^{n+1} Y & \stackrel{\Phi[n+1]}{\to} & Y D^{n+1} \\ & _\mathllap{id} \searrow & \downarrow_\mathrlap{m T^{n-1} Y} & & \downarrow_\mathrlap{Y\epsilon D^n} \\ & & T^n Y & \underset{\Phi[n]}{\to} & Y D^n. }

      Since all the maps are algebra maps and uT nY:T nYT n+1Yu T^n Y: T^n Y \to T^{n+1} Y exhibits T n+1YT^{n+1} Y as the free algebra on T nYT^n Y, it suffices to check commutativity around the perimeter. (N.B.: the triangle commutes, even in the case where n=0n=0 which we need to start the induction.) By definition of Φ[n+1]\Phi[n+1], commutativity of the perimeter boils down to commutativity of

      T nY uT nY T n+1Y Φ[n+1] YD n+1 Φ[n] hD n YϵD n YD n id YD n\array{ T^n Y & \stackrel{u T^n Y}{\to} & T^{n+1} Y & \stackrel{\Phi[n+1]}{\to} & Y D^{n+1} \\ & _\mathllap{\Phi[n]} \searrow & & \nearrow_\mathrlap{h D^n} & \downarrow_\mathrlap{Y\epsilon D^n} \\ & & Y D^n & \underset{id}{\to} & Y D^n }

      where the triangle commutes by one of the acyclic structure equations.

    • For j1j \geq 1, the commutativity of the right square in
      T nY uT nY T n+1Y PhiY[n+1] YD n+1 T j1mT nj1Y nat. T jmT nj1Y YD jϵD nj T n1Y uT n1Y T nY Φ[n] YD n Φ[n1] hD n1 YD n1 \array{ T^n Y & \stackrel{u T^n Y}{\to} & T^{n+1} Y & \stackrel{\PhiY[n+1]}{\to} & Y D^{n+1} \\ _\mathllap{T^{j-1}m T^{n-j-1} Y} \downarrow & nat. & \downarrow_\mathrlap{T^j m T^{n-j-1} Y} & & \downarrow_\mathrlap{Y D^j \epsilon D^{n-j}} \\ T^{n-1} Y & \underset{u T^{n-1} Y}{\to} & T^n Y & \underset{\Phi[n]}{\to} & Y D^n \\ & _\mathllap{\Phi[n-1]} \searrow & & \nearrow_\mathrlap{h D^{n-1}} & \\ & & Y D^{n-1} & & }

      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 Φ[n]\Phi[n]. But the perimeter commutes by examining the diagram

      T nY uT nY T n+1Y Φ[n+1] YD n+1 T j1mT nj1Y Φ[n] hD n YD jϵD nj T n1Y ind. YD n nat. YD n Φ[n1] hD n1 YD n1 \array{ T^n Y & \stackrel{u T^n Y}{\to} & T^{n+1} Y & \stackrel{\Phi[n+1]}{\to} & Y D^{n+1} \\ _\mathllap{T^{j-1}m T^{n-j-1} Y} \downarrow & _\mathllap{\Phi[n]} \searrow & & \nearrow_\mathrlap{h D^n} & \downarrow_\mathrlap{Y D^j \epsilon D^{n-j}} \\ T^{n-1} Y & ind. & Y D^n & nat. & Y D^n \\ & _\mathllap{\Phi[n-1]} \searrow & \downarrow & \nearrow_\mathrlap{h D^{n-1}} & \\ & & Y D^{n-1} & & }

      (where the middle vertical arrow is YD j1ϵD njY D^{j-1} \epsilon D^{n-j}) using the inductive hypothesis in the bottom left parallelogram.

  • To check preservation of degeneracy maps, we treat separately the cases j=1j=1 and j2j \geq 2.

    • For j=1j = 1, the commutativity of the top right square in
      T n1Y uT n1Y T nY Φ[n] YD n uT n1Y nat. TuT n1Y YδD n1 T nY uT nY T n+1Y Φ[n+1] YD n+1 Φ[n] hD n YD n \array{ T^{n-1} Y & \stackrel{u T^{n-1} Y}{\to} & T^n Y & \stackrel{\Phi[n]}{\to} & Y D^n \\ _\mathllap{u T^{n-1} Y} \downarrow & nat. & \downarrow _\mathrlap{T u T^{n-1} Y} & & \downarrow_\mathrlap{Y \delta D^{n-1}} \\ T^n Y & \underset{u T^n Y}{\to} & T^{n+1} Y & \underset{\Phi[n+1]}{\to} & Y D^{n+1} \\ & _\mathllap{\Phi[n]} \searrow & & \nearrow_\mathrlap{h D^n} & \\ & & Y D^n & & }

      is by appeal to a freeness argument where we just need to check commutativity of the perimeter (the special case n=1n=1 being used to start the induction). But this boils down to commutativity of the diagram

      T n1Y uT n1Y T nY Φ[n] YD n uT n1Y Φ[n1] hD n1 YδD n1 T nY YD n1 YD n+1 Φ[n] hD n1 hD n YD n \array{ T^{n-1} Y & \stackrel{u T^{n-1} Y}{\to} & T^n Y & \stackrel{\Phi[n]}{\to} & Y D^n \\ _\mathllap{u T^{n-1} Y} \downarrow & \searrow_\mathrlap{\Phi[n-1]} & & _\mathllap{h D^{n-1}} \nearrow & \downarrow_\mathrlap{Y \delta D^{n-1}} \\ T^n Y & & Y D^{n-1} & & Y D^{n+1} \\ & _\mathllap{\Phi[n]} \searrow & \downarrow_\mathrlap{h D^{n-1}} & \nearrow_\mathrlap{h D^n} & \\ & & Y D^n & & }

      where the bottom right quadrilateral commutes by one of the acyclic structure equations.

    • For j2j \geq 2, the commutativity of the top right square in
      T n1Y uT n1Y T nY Φ[n] YD n T j1uT njY nat. T juT njY YD j1δD nj T nY uT nY T n+1Y Φ[n+1] YD n+1 Φ[n] hD n YD n \array{ T^{n-1} Y & \stackrel{u T^{n-1} Y}{\to} & T^n Y & \stackrel{\Phi[n]}{\to} & Y D^n \\ _\mathllap{T^{j-1} u T^{n-j} Y} \downarrow & nat. & \downarrow _\mathrlap{T^j u T^{n-j} Y} & & \downarrow_\mathrlap{Y D^{j-1} \delta D^{n-j}} \\ T^n Y & \underset{u T^n Y}{\to} & T^{n+1} Y & \underset{\Phi[n+1]}{\to} & Y D^{n+1} \\ & _\mathllap{\Phi[n]} \searrow & & \nearrow_\mathrlap{h D^n} & \\ & & Y D^n & & }

      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

      T n1Y uT n1Y T nY Φ[n] YD n T j1uT njY Φ[n1] hD n1 YD j1δD nj T nY ind. YD n1 nat. YD n+1 Φ[n] hD n YD n \array{ T^{n-1} Y & \stackrel{u T^{n-1} Y}{\to} & T^n Y & \stackrel{\Phi[n]}{\to} & Y D^n \\ _\mathllap{T^{j-1} u T^{n-j} Y} \downarrow & \searrow_\mathrlap{\Phi[n-1]} & & _\mathllap{h D^{n-1}} \nearrow & \downarrow_\mathrlap{Y D^{j-1} \delta D^{n-j}} \\ T^n Y & ind. & Y D^{n-1} & nat. & Y D^{n+1} \\ & _\mathllap{\Phi[n]} \searrow & \downarrow & \nearrow_\mathrlap{h D^n} & \\ & & Y D^n & & }

      where the middle vertical arrow is YD j2δD njY D^{j-2} \delta D^{n-j}.

This completes the proof.

Special cases

The classifying space and universal bundle of a group

There is a monad TT on Set Set whose algebras are left G G -sets, given by

TX=G×X. T X \,=\, G \times X \,.

The terminal GG-set 11, namely the singleton set equipped with the trivial action of GG, is an algebra for this monad TT.

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 Bar T(1)Bar_T(1) for the augmented simplicial set obtained by applying this to 1GSet1 \in G Set.

By Theorem , its underlying simplicial set UBar T(1)U Bar_T(1) is contractible. This simplicial set is often denoted EGE G (and as such known as the universal simplicial GG-principal bundle) or WGW G (see here at simplicial classifying space).

Working through the details, we see that the set of nn-simplices of EGE G is G n+1G^{n+1}, and all the face maps i:G n+1G n\partial_i \colon G^{n+1} \to G^n are given by multiplying successive entries in the (n+1)(n+1)-tuple, except for the last face map, which encodes the trivial action of GG on 11:

i(g 0,g 1,,g n)=(g 0,,g ig i+1,,g n) \partial_i (g_0, g_1, \dots, g_n) \;=\; (g_0, \dots, g_i g_{i+1}, \dots, g_n)

for i=0,,n1i = 0, \dots, n-1, while

n(g 0,g 1,,g n)=(g 0,g 1,,g n1). \partial_n (g_0, g_1, \dots, g_n) \;=\; (g_0, g_1, \dots, g_{n-1}) \,.

The quotient BG(EG)/GB G \coloneqq (E G)/G is known as the simplicial classifying space of GG (also denoted W¯G\overline{W}G), and its topological realization is a topological classifying space for GG-principal bundles. Finally the topological realization of the quotient map EGBGE G \to B G is the “universal GG-principal bundle”.

The simplicial set BGB G is also the simplicial nerve of the delooping groupoid of GG.

For modules over an algebra

Let AA be a commutative associative algebras over some ring kk. Write AModA Mod for the category of right modules over AA.

For MM a right module, also M kAM \otimes_k A is canonically a right module. This construction extends to a functor

T=() kA:AModAMod. T = (-) \otimes_k A : A Mod \to A Mod \,.

The monoid structure on AA makes TT into a monad on AModA Mod: the monad product and unit are given by the product and unit in AA.

For any right AA-module MM, the action ρ:MAN\rho \colon M \otimes A \to N makes MM into an algebra of the monad TT. The bar construction Bar T(A)Bar_T(A) is then the simplicial AA-module

N kA kAρIdIdμN kA. \cdots \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} N \otimes_k A \otimes_k A \stackrel{\overset{Id \otimes \mu}{\longrightarrow}}{\underset{\rho \otimes Id}{\longrightarrow}} N \otimes_k A \,.

Under the Moore complex functor of the Dold-Kan correspondence, this simplicial AA-module is identified with a chain complex of AA-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 AA, which can be used to compute the Tor group

Tor(M,A×A). Tor(M, A \times A) \,.

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

For differential graded (Hopf) algebras

See bar and cobar construction.

For E E_\infty-algebras

See (Fresse).

References and Literature

The original reference for bar constructions in the generality of monads is

  • Roger Godement, Topologie algébrique et theorie des faisceaux, Actualités Sci. Ind. 1252, Hermann, Paris (1958) [webpage, pdf]

A general discussion of bar construction for monads is at

Textbook accounts can be found 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

The compositional structure of the bar construction of several monads, as well as its interpretation in terms of partial evaluations is studied in

  1. N.B.: including the empty ordinal.

  2. If X:Δ a opCX: \Delta_a^{op} \to \mathbf{C} is a simplicial object, then X([n])X([n]) is what is usually denoted X n1X_{n-1}, the object of cells in dimension n1n-1. Note that X([0])=X 1X([0]) = X_{-1} is the augmented component. The nn can be thought of as the number of vertices of a simplex of dimension n1n-1. We choose the index nn over the geometric dimension n1n-1 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.