Bar Constructions
Todd Trimble
September 5, 1999
This is part B of a long email from Todd Trimble; part
A is here. Both parts have been
nicely LaTeXed by Samuel Vidal:
and the LaTeX files are here.
B. Bar constructions
In this section, we initially work over a base topos Set; all of our
constructions apply in a context where we work over a Grothendieck
topos E as base. Later in this section, we take E to be the topos
of Joyal species.
Classically, bar constructions have been used to build classifying
bundles, free resolutions for group cohomology, and similar constructs.
A common characteristic between these constructions is the production of
an acyclic algebraic structure (e.g., a contractible G-space EG, or
a free resolution of a G-module). We begin by formalizing this
characteristic in terms of a universal property for bar constructions.
B.1. Acyclic structures
We follow the algebraists' convention, taking the simplicial category
to mean the category of finite ordinals (including the empty ordinal)
and order-preserving maps. It is well-known that \Delta is initial
amongst strict monoidal categories equipped with a monoid, which in
\Delta is 1. This induces a monad 1 + - on \Delta, called the translation
monad. By composition, this in turn induces a pullback comonad P on simplicial
sets; by the Kan construction, it has a left adjoint which is a monad C
on simplicial sets, called the cone monad.
To explain this terminology, we recall the topologists' convention, where
\Delta is restricted to the full subcategory \Delta_+ of non-empty ordinals.
If S-set denotes the category of simplicial sets under the algebraists'
convention, and S_{+}-set that under the topologists' convention, then
by restriction we get a functor S-set --> S_{+}-set, which has a left
adjoint. The left adjoint augments a S_{+} set X by its set of path
components \pi_{0}(X). Starting with X, we can apply the left augmentation,
followed by the cone monad, followed by restriction S-set --> S_{+} set:
this gives a monad which, passing to geometric realization, is the
mapping cone of X --> \pi_{0}(X). The right adjoint to this monad carries
a comonad structure; the category of algebras over the monad is equivalent
to the category of coalgebras over the comonad. This category of
algebras/coalgebras could be called the "acyclic topos": the algebras X
are acyclic as simplicial sets. More to the point, the algebra structure
CX --> X witnesses this acyclicity by providing a representative basepoint
for each path component of X, together with a well-behaved simplicial
homotopy which contracts each component down to its basepoint.
Definition: An acyclic structure is an algebra over C (or coalgebra over P).
An S-acyclic structure is one augmented over a set S.
A morphism between acyclic structures is just a C-algebra map; a morphism
between S-acyclic structures is one whose component at S is the identity.
It doesn't matter whether the monad C is taken under the algebraists' or
topologists' convention: the category of C-algebras in S-set is equivalent to
the category of C-algebras in S_{+}-set, since given a C-algebra in S-set,
-->
... X_{1} --> X_{0} --> X_{-1},
it follows from acyclicity that this portion of the simplicial structure
is a split coequalizer, so that the augmentation map is the usual
augmentation to its set of path components.
In the sequel, it will be useful to regard an acyclic structure,
as a functor X: \Delta^{op} --> Set, as a right coalgebra XT --> X
over the translation comonad T: \Delta^{op} --> \Delta^{op}.
B.2. Abstract bar constructions
Next, let us recall the formalism which leads to bar constructions.
Let U: A --> E be a monadic functor over a category E (in most of
our applications, E will be a topos). Let F: E --> A be the left
adjoint, so that we have a monad M: E --> E and a comonad C: A --> A.
The comonad may be regarded as a comonoid in the endofunctor category
[A, A] (under the monoidal product given by endofunctor composition).
Since \Delta^{op} is initial amongst strict monoidal categories
equipped with a comonoid, there exists a unique monoidal functor
\Delta^{op} --> [A, A]
sending the comonoid 1 to the comonoid C. Now if X is an object of
A (i.e., an M-algebra), we have an evaluation functor ev_{X}: [A, A] --> A.
Thus we have the following composition which leads to a simplicial
E-object:
ev_{X} U
\Delta^{op} --> [A, A] -----> A --> E
and this defines what we mean by the bar construction, denoted B(M, M, X).
More explicitly, this is an augmented simplicial object of the form
\muX
--->
... MMX ---> MX --> X
M\xi \xi
and this simplicial object admits an acyclic structure where the contracting
homotopy is built out of components of the unit u of the monad M:
uM^{n}X
M^{n}X -------> M^{n+1}X.
(The "co-associativity" axiom for the right T-coalgebra structure holds
by naturality of u.)
Definition: An X-acyclic M-algebra is a simplicial M-algebra whose
underlying simplicial E-object admits an X-acyclic structure.
Notice that we do not require any compatibility between the M-algebra
structure and the acyclic structure: the acyclic structure map is not
required, for example, to be a map of simplicial M-algebras.
A morphism of X-acyclic M-algebras is just a map of simplicial M-algebras
whose underlying simplicial E-map is a morphism of X-acyclic structures.
The category of X-acyclic M-algebras may thus be rendered as a (2-)pullback
of
X-Acyclic(E)
|
| "forget"
V
[\Delta^{op}, A] ---------> [\Delta^{op}, E]
[1, U]
where the vertical arrow is the obvious forgetful functor from the
category of X-acyclic structures in E.
The universal property of the bar construction is enunciated in the following
Theorem: B(M, M, X) is initial in the category of X-acyclic M-algebras.
Proof: The method of proof closely parallels that of the acyclic
models theorem familiar from homological algebra. Let Y be
an acyclic M-algebra augmented over X:
-->
... Y_{1} --> Y_{0} --> X
so that the T-coalgebra structure or contracting homotopy has
the form X --> Y_{0} --> Y_{1} --> ... In order to get the desired
simplicial M-algebra map B(M, M, X) --> Y which preserves the
contracting homotopies, we are forced to use the diagram
uX uMX uMMX
X --> MX ---> MMX ---> MMMX ---> ...
| | | |
1 | |\phi_1 |\phi_2 |\phi_3 ...
| | | |
V V V V
X --> Y_0 --> Y_1 ---> Y_2 ----> ...
h_0 h_1 h_2
where \phi_{n+1} is defined as the unique M-algebra map which
extends (h_n)(\phi_{n}) along uM^{n}X. Thus uniqueness is clear;
what remains is to check that the \phi_n form the components of
a simplicial map \phi. The proof of this is sketched in the
lemma and corollary which follow.
Let Simp(E) denote the category of simplicial objects in E. The pullback
comonad P may be regarded as a comonoid in the endofunctor category
[Simp(E), Simp(E)], so that there is an induced monoidal functor
\Delta^{op} --> [Simp(E), Simp(E)]
which we may compose with evaluation at a simplicial object Y. If
moreover Y carries an acyclic structure, then there is an induced acyclic
structure on this composite, regarded as a simplicial object in Simp(E):
ev_Y
\Delta^{op} --> [Simp(E), Simp(E)] ----> Simp(E)
We denote this bisimplicial object by B(Y, T, T), where T is the aforesaid
translation comonad. Since PY = YT, it has the form
-->
... YTT --> YT --> Y.
Observe that applying evaluation Simp(E) --> E at the augmented component
(the component we had earlier indexed by the numeral -1), this yields Y again:
-->
... Y_1 --> Y_0 --> Y_{-1} = X.
Lemma: Let Y be an acyclic M-algebra. Let B(M, M, Y) be the bisimplicial
object formed as the bar construction on Y. Then there exists a
map of Y-acyclic M-algebras (valued in Simp(E))
B(M, M, Y) --> B(Y, T, T)
whose value at the terminal object 0 in \Delta^{op} is the identity
map on Y.
Thinking of B(M, M, Y) --> B(T, T, Y) as a transformation between functors
of the form Delta^{op} --> Simp(E), we may post-compose by evaluation
Simp(E) --> E at the augmented component. This yields precisely
B(M, M, X) --> Y
whence follows a corollary which completes the proof of the theorem:
Corollary: There exists an X-acyclic M-algebra map B(M, M, X) --> Y.
Proof of lemma: We construct B(M, M, Y) --> B(Y, T, T) inductively, beginning
with the identity on Y. The next component is of the form
\theta_{1}: MY --> YT, namely the composite
Mh \xi T
MY --> MYT -----> YT
where h is the right T-coalgebra structure and \xi the
M-algebra structure. It is immediate that \theta_{1}
preserves the homotopy component (i.e., h = \theta_{1}(uY)),
and since \theta_{1} is an M-algebra map, it quickly
follows that (Y\ep)\theta_{1} = \xi, where \ep is the
counit of the comonad T.
We leave to the reader to check that if we inductively
define \theta_{n}: M^{n}Y --> YT^{n} as the composite
M^{1}\theta_{n-1} \theta_{1}T^{n-1}
M^{n}Y ----------------> MYT^{n-1} ----------------> YT^{n}
then it easily follows that
(\theta_{n})(uM^{n-1}Y) = (hT^{n-1})(\theta_{n-1}), i.e.,
\theta preserves the homotopies (preserves acyclic structure).
The fact that \theta_{n} preserves face and degeneracy maps
follows by induction: since \theta_{n} is an M-algebra map
by construction, it suffices to check that the relevant
diagrams which obtain by precomposing with a unit u commute,
but since \theta preserves homotopies, one can exploit the
naturality of the homotopies to convert the diagrams into
ones where the inductive assumption applies. In short, the
argument is similar to the usual one for the acyclic models
theorem, and this completes our sketch of the proof.
B.3. Applications
A classical application of bar constructions is the Milgram bar construction
of a classifying bundle (say of a discrete group G). As is well known, the
total space EG is a contractible space on which G acts freely. What appears
less well known is the following theorem.
Let us define a contractive space to be a space (in a suitable topological
category, such as the category of compactly generated Hausdorff spaces)
which is an algebra over the cone monad. Here, the cone monad means
the mapping cone of the map X --> 1 into the one-point space, and this
is the monad whose algebras are pointed spaces equipped with a continuous
action by the unit interval I, the monoid whose multiplication is "inf",
such that multiplication by 0 sends every point to the basepoint. An
algebra structure may be viewed as a well-behaved homotopy which contracts
the space to a point.
Theorem: EG is initial amongst G-spaces whose underlying space is equipped
with a contractive structure.
Proof: Let R: S-set --> Top be geometric realization. EG is formed as
RB(G, G, 1), where the bar construction is applied to the monad
G x - on Set. This is a 1-acyclic space, i.e., a contractive space.
If X is any other contractive G-space, we wish to demonstrate that
there is exactly one contractive G-map EG --> X.
If S: Top --> S-set is the singularization functor right adjoint to
R, then a map EG --> X gives rise to B(G, G, 1) --> SX. Since G x -
as a functor Top --> Top is cocontinuous, it is easy to see that
a G-map EG --> X gives rise to a G-map B(G, G, 1) --> SX. Next, let
C be the cone monad acting on the category of pointed spaces; then
C is also cocontinuous (it has a right adjoint given by the path
space functor), and it follows as before that a C-map EG --> X gives
rise to a C-map B(G, G, 1) --> SX. Indeed, contractive G-maps
EG --> X are in bijective correspondence with 1-acyclic G-maps
B(G, G, 1) --> SX in S-set, and there is exactly one of these by
the theorem of the last section. The proof is complete.
Now let E be the topos of Joyal species, [P, Set], where P denotes the
permutation category. If Op denotes the category of permutative operads,
then the underlying functor Op --> [P, Set] is monadic. Let O denote
the monad for this adjunction. If M is an O-algebra (an operad), then
there is an associated bar construction B(O, O, M). Needless to say,
it is acyclic.
Example 1: (Associahedra revisited) Let t_+ be empty in degree 0, and
terminal in higher degrees. There is a unique operad structure
on t_+, as a suboperad of the terminal operad t.
Form the bar construction B(O, O, t_{+}): this carries a
simplicial O-algebra structure, i.e., a simplicial operad
structure. If we form the operad quotient in which every
unary operation is identified with the identity operation,
then the operad which results is a permutative version of the
associahedral operad given in part A.
Example 2: Now let t be the terminal operad, and form the simplicial
operad B(O, O, t). Again take the operad quotient in which
the operation in t of degree 1 is identified with the operad
identity. The result is a simplicial operad called the
"monoidahedral operad", denoted M.
The principle is that we have included a generating operation
in degree 0, so that we obtain not just higher associativity
laws as in the associahedral operad, but higher unit laws as
well. In contrast to the associahedral operad, the nerve
components here are infinite-dimensional (again, due to the
presence of a nullary operation).
This last example has interesting applications to Trimble's work on
Grothendieck's fundamental n-groupoids. Working in the category of
bipointed spaces and bipointed maps, let I denote the unit interval,
bipointed by the pair (0, 1). On the category of bipointed spaces, there
is a monoidal product given by "wedges" X v Y, where the second point of
X is identified with the first point of Y, and the wedge is bipointed by
the first point of X followed by the second point of Y. Then the n-fold
wedge of I is canonically identified with the interval [0, n], bipointed
by the pair (0, n).
Now suppose we adapt the monoidahedral operad to the non-permutative setting.
Let Bip(I, I^n) denote the space of bipointed maps from I to its n-fold
wedge; this is the n-th component of a tautological non-permutative
operad structure. Notice that these spaces are convex, so that if we take
as basepoint in Bip(I, I^n) the map I --> I^n given by multiplication by n,
then by convexity there is an induced contractive structure on each of these
spaces. Thus we get a contractive spatial operad Bip(I, I^*).
Then there is a 1-acyclic simplicial operad S(Bip(I, I^*)), obtained by
applying singularization to the aforesaid spatial operad. It follows that
there is a unique 1-acyclic operad map M --> S(Bip(I, I^*)). This map is
used to identify higher associativity and higher unit laws present in
Bip(I, I^*).
Before giving the next few examples (which are relevant to weak n-functors
and their geometry), we need a new definition. If X and Y are species,
we use X.Y to denote their substitution product.
Definition: Let A and B be operads. An A-B bimodule is a species X equipped
with structure maps A.X --> X and X.B --> X, compatible in the
usual sense.
It is tempting to try to define bimodule composition, where if X is an
A-B bimodule and Y is a B-C bimodule, then the tensor product XY is an
A-C bimodule given by an evident coequalizer of the form
-->
X.B.Y --> X.Y -->> XY
The trouble is that bimodule composition fails to be associative, because
while -.Y preserves colimits, X.- does not. However, in practice many
coequalizers of the type shown above split, and we can refer to triple
products XYZ without essential ambiguity if the coequalizers ending with
XY and YZ split.
If M is an operad, then the free M-bimodule monad F is given by the
assignment X |--> M.X.M, and if X is itself a bimodule, we obtain a
bar construction B(F, F, X). Some examples follow.
Example 3: Let t_+ be the permutative version of the operad of example 1,
viewed as a bimodule over itself. Then B(F, F, t_{+}) is a
contractive t-bimodule whose components are triangularized
*cubes*.
Let us calculate this in detail. In the language of Joyal species and
their analytic functors, we have the linear fractional transformation
t_{+}(X) = X/(1-X).
Let t_{+}^n denote the n-fold substitution power of t_{+}. Then the
bar construction
-->
... t_{+}^5 --> t_{+}^3 --> t_{+}
has the form -->
... X/(1-5X) --> X/(1-3X) --> X/(1-X)
Looking at coefficients, a structure of species X/(1-kX) on n points
is given by a "combing" or linear order on {1, ..., n} together with
a function {1, ..., n} --> {1, ..., k}. We abbreviate the set of such
structures by k^n. Keeping the object n of the permutation category P
fixed, we get a simplicial object
--> -->
(... 5^n --> 3^n --> 1) = (... 5 --> 3 --> 1)^n
where the latter power denotes n-fold cartesian product in S-set.
Passing now to the topologists' convention (i.e., truncating the
augmented object 1), the claim is that
--> -->
... 7 --> 5 --> 3
-->
is the nerve of a once-subdivided 1-cube : .___.___.
Certainly there are 3 0-cells, and 5 minus 3 non-degenerate 1-cells.
If n_k denotes the number of non-degenerate k-cells, then we have
1.(n_0) = 3
1.(n_1) + 1.(n_0) = 5
1.(n_2) + 2.(n_1) + 1.(n_0) = 7
etc. in Pascal triangle fashion, so that n_k = 0 for k>1. It is then
very easy to demonstrate that we thus in fact get a subdivided 1-cube
for the component n=1. For higher n, we take powers of this 1-cube, and
this leads to an n-cube, suitably triangularized.
Example 4: Let A be the associahedral operad of example 1, and consider
t_{+} as an A-A bimodule. Then the bar construction B(F, F, t_{+})
gives a canonical triangulation of the polyhedra which
parametrize the operations of Stasheff's A_n maps (see
Homotopy Associativity of H-Spaces I, II).
This example deserves further comments. There is a strong analogy between
the cellular structure of the A_n maps, and the cellular structure of the
data for bihomomorphisms, trihomomorphisms, etc., except that the A_n
structures and A_n maps take account only of higher associativities and
their weak preservations, but do not take account of units.
To take account of units, the geometry of A_n maps should be replaced by
the geometry of the bar construction B(F, F, t), where the terminal operad
t is regarded as a bimodule over the monoidahedral operad M of example 2.
The polyhedra which result are again infinite-dimensional. I call these
polyhedra "functoriahedra"; the claim/conjecture is that they carry all
the cellular structure one desires of weak n-functors.
© 1999 Todd H. Trimble