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