nLab
two-sided bar construction

Contents

Idea

A two-sided bar construction is a simplicial object associated with the data of a left module and right module over a monad.

Definition

Left and right modules

Let B\mathbf{B} be a 2-category, and let M:BBM: B \to B be a monad with multiplication m:MMMm: M M \to M and unit u:1 BMu: 1_B \to M. Recall that a left module over MM consists of a 1-cell X:ABX: A \to B and a 2-cell α:MXX\alpha: M X \to X such that the diagrams

MMX Mα MX X 1 X X mX α uX 1 X MX α X MX α X\array{ M M X & \stackrel{M\alpha}{\to} & M X & & & X & \stackrel{1_X}{\to} & X \\ m X \downarrow & & \downarrow \alpha & & & u X \downarrow & & \downarrow 1_X\\ M X & \underset{\alpha}{\to} & X & & & M X & \underset{\alpha}{\to} & X }

commute. A right module over MM consists of a 1-cell Y:BCY: B \to C and a 2-cell β:YMY\beta: Y M \to Y such that the diagrams

YMM βM YM Y 1 Y Y Ym β Yu 1 Y YM β Y YM β Y\array{ Y M M & \stackrel{\beta M}{\to} & Y M & & & Y & \stackrel{1_Y}{\to} & Y \\ Y m \downarrow & & \downarrow \beta & & & Y u \downarrow & & \downarrow 1_Y\\ Y M & \underset{\beta}{\to} & Y & & & Y M & \underset{\beta}{\to} & Y }

commute.

Initial structure

With the aid of string diagrams, one may easily visualize the structure I\mathbf{I} that is initial amongst 2-categories equipped with the data of monad and left and right modules over the monad. The 2-category has three 0-cells 00, 11, and 22, 1-cells which are of the form

m n:11ym n:12m nx:01ym nx:02,m^{\circ n}: 1 \to 1 \qquad y m^{\circ n}: 1 \to 2 \qquad m^{\circ n} x: 0 \to 1 \qquad y m^{\circ n} x: 0 \to 2,

and the nonempty hom-categories hom(i,j)=I(i,j)hom(i, j) = \mathbf{I}(i, j) may be described as follows:

  • hom(1,1)hom(1, 1) is isomorphic to the “algebraist’s Δ a\Delta_a” (see simplex category): the category of finite ordinals and order-preserving maps;

  • hom(1,2)hom(1, 2) is isomorphic to Δ \Delta_{\bot}: the category of nonempty finite ordinals and order-preserving maps that preserve the bottom element;

  • hom(0,1)hom(0, 1) is isomorphic to Δ \Delta_{\top}: the category of nonempty finite ordinals and order-preserving maps that preserve the top element;

  • hom(0,2)hom(0, 2) is isomorphic to the “topologist’s Δ op\Delta^{op}” (the usual simplex category), which is isomorphic to the category Δ ,\Delta_{\top, \bottom} of finite intervals, whose objects are finite ordinals with at least two elements, and whose morphisms are order-preserving maps that preserve both top and bottom.

Note that the structure of hom(1,1)hom(1, 1) simply recapitulates the universal property of the algebraist’s Δ a\Delta_a, as initial among strict monoidal categories equipped with a monoid MM. This result is most easily understood in terms of string diagrams, and the structure of the other hom-categories is guided by similar string diagram considerations.

Two-sided bar construction

Now we give the definition. Suppose given a 2-category B\mathbf{B} together with a monad M:BBM: B \to B in B\mathbf{B}, together with a left module X:ABX: A \to B and a right module Y:BCY: B \to C. There is a unique 2-functor

IB\mathbf{I} \to \mathbf{B}

which preserves the monad and module structures, and this induces a functor

Δ opI(0,2)B(A,C)\Delta^{op} \cong \mathbf{I}(0, 2) \to \mathbf{B}(A, C)

This functor is the two-sided bar construction, denoted B(Y,M,X)B(Y, M, X).

The structure of the two-sided bar construction may be given more concretely as follows:

  • The nn-dimensional component of B(Y,M,X)B(Y, M, X) is

    B(Y,M,X) n=YM nXB(Y, M, X)_n = Y M^{\circ n} X
  • The n+2n+2 face maps

    d n i:YM n+1XYM nXd_{n}^{i}: Y M^{n+1} X \to Y M^n X

    are βM nX\beta M^n X (i=0i = 0), YM i1mM niXY M^{i-1} m M^{n-i} X (1in1 \leq i \leq n), and YM nαY M^n \alpha (i=n+1i = n+1).

  • The n+1n+1 degeneracy maps

    s n i:YM nXYM n+1Xs_{n}^{i}: Y M^n X \to Y M^{n+1} X

    are YM iuM niXY M^i u M^{n-i} X (0in0 \leq i \leq n).

Examples

Two-sided bar constructions encapsulate a surprising number of constructions, a few of which follow.

Classifying bundle

Consider the cartesian monoidal category TopTop as a 1-object bicategory ΣTop\Sigma Top (which we may strictify to a 2-category). A topological monoid MM is the same as a monad in ΣTop\Sigma Top, and the usual meaning of left and right MM-modules is preserved by thinking of them as modules over the monad.

In particular, MM may be regarded as a left or right MM-module, and the 1-point space 11 carries a unique structure of left or right MM-module. As a result we may consider the simplicial space

BM=B(1,M,1)B M = B(1, M, 1)

as base space, and the simplicial space

EM=B(M,M,1)E M = B(M, M, 1)

as total space, of a simplicial fibration

B(π,M,1):B(M,M,1)B(1,M,1)B(\pi, M, 1): B(M, M, 1) \to B(1, M, 1)

induced by the unique left module map π:M1\pi: M \to 1. This is the classifying bundle? of the monoid MM.

Cofibrant replacement

If MM is a monad and (X,α:MXX)(X, \alpha: M X \to X) is a (left-sided) MM-algebra, then with MM acting upon itself on the right, there is a simplicial object

B(M,M,X)B(M, M, X)

which may be regarded as a cofibrant replacement of XX, a simplicial MM-algebra which as a simplicial object is homotopy-equivalent to the constant simplicial object at XX. (More should be added here.) See also bar construction.

Rather more generally, if in addition (Y,β:YMY(Y, \beta: Y M \to Y is a right MM-module, we may denote the coequalizer of the pair

YMXYαβXYXY M X \stackrel{\overset{\beta X}{\to}}{\underset{Y \alpha}{\to}} Y X

(if it exists) by Y MXY \circ_M X, and think of it as the tensor product (over MM) of YY and XX. The general principle is that the simplicial object B(Y,M,X)B(Y, M, X) is canonically augmented by Y MXY \circ_M X, and serves as a cofibrant replacement of Y MXY \circ_M X when the latter is regarded as a constant simplicial object. See also homotopy colimits, below.

Canonical two-sided bar construction of an adjunction

Suppose given any adjoint pair (F:BA)(U:AB)(F: B \to A) \dashv (U: A \to B) in a 2-category B\mathbf{B}, with counit ε:FU1 A\varepsilon: F U \to 1_A. There is an associated monad M=UF:BBM = U F: B \to B, and a canonical left MM-action on UU:

α=Uε:UFUU,\alpha = U \varepsilon: U F U \to U,

and a canonical right MM-action on FF:

β=εF:FUFF.\beta = \varepsilon F: F U F \to F.

We may then form the canonical simplicial object B(F,M,U)B(F, M, U). By general abstract nonsense, the tensor product F MUF \circ_M U is 1 A1_A, so if we regard 1 A1_A as a constant simplicial object Δ opB(A,A)\Delta^{op} \to \mathbf{B}(A, A), the cofibrant replacement result above specializes as follows.

Proposition

The canonical simplicial map B(F,M,U)1 AB(F, M, U) \to 1_A is a simplicial homotopy equivalence.

Delooping machines

The classic application of this two-sided bar construction was given by Peter May, after work of Jon Beck. Let Ω=Top *(S 1,):Top *Top *\Omega = Top_{*}(S^1, -): Top_{*} \to Top_{*} be the loop space functor on pointed spaces, with left adjoint the suspension functor S=S 1:Top *Top *S = S^1 \wedge -: Top_{*} \to Top_{*}. For each n1n \geq 1, the adjunction S nΩ nS^n \dashv \Omega^n induces a monad Ω nS n\Omega^n S^n which acts (on the left) on nn-fold loop spaces Ω nX\Omega^n X, and acts on the right on S nS^n.

Proposition

The canonical simplicial map B(S n,Ω nS n,Ω nX)XB(S^n, \Omega^n S^n, \Omega^n X) \to X is a simplicial homotopy equivalence.

So if YY is an nn-fold loop space, the two-sided bar construction B(S n,Ω nS n,Y)B(S^n, \Omega^n S^n, Y) provides its nn-fold delooping.

Continuing this train of thought: suppose given a monad C nC_n equipped with a monad map ϕ:C nΩ nS n\phi: C_n \to \Omega^n S^n (which is mated to an action C nΩ nΩ nC_n \Omega^n \to \Omega^n) such that ϕ\phi is objectwise a homotopy equivalence. Then ϕ\phi induces a homotopy equivalence

B(S n,ϕ,Ω n):B(S n,C n,Ω n)B(S n,Ω nS n,Ω n)B(S^n, \phi, \Omega^n): B(S^n, C_n, \Omega^n) \to B(S^n, \Omega^n S^n, \Omega^n)

The classic example of this is where C nC_n is the little nn-cubes operad, which acts canonically on Ω n=Top *(S n,)\Omega^n = Top_{*}(S^n, -). The advantage is that the monad C nC_n is much more tractable than Ω nS n\Omega^n S^n.

Continuing this thought still further, we have the following recognition theorem of May:

Theorem

Suppose YY is a connected space with a C nC_n-action. Then YY is homotopy equivalent to an nn-fold loop space, and indeed there is a canonical span

Ω nB(S n,C n,Y)B(Ω nS n,C n,Y)B(Ω nS n,Ω nS n,Y)Y\Omega^n B(S^n, C_n, Y) \leftarrow B(\Omega^n S^n, C_n, Y) \to B(\Omega^n S^n, \Omega^n S^n, Y) \simeq Y

where each of the arrows is a homotopy equivalence.

I’m going on sheer memory here of May’s Geometry of Iterated Loop Spaces, and it’s quite possible that I’m missing hypotheses in some directions and that hypotheses could be weakened in other directions, e.g., replacing the connectedness assumption with a group completion assumption. This is in need of expert attention.

Homotopy colimits

Suppose that CC is a small category and F:CTopF: C \to Top is a functor. We may regard CC as a monad C:C 0C 0C: C_0 \to C_0 in the bicategory of spans in TopTop, where C 0C_0 is the set of objects with the discrete category, and we may regard FF as a left module over the monad CC.

As always, the terminal object 11 carries a unique right module structure. The usual colimit, colimFcolim F, may be described as the tensor product

colimF1 CFcolim F \cong 1 \circ_C F

As a result, we have the cofibrant replacement B(1,C,F)B(1, C, F) of colimFcolim F. The geometric realization of the simplicial space B(1,C,F)B(1, C, F) is none other than the homotopy colimit of FF.

Revised on June 14, 2010 10:00:49 by Urs Schreiber (131.211.36.96)