symmetric monoidal (∞,1)-category of spectra
The bar construction takes a monad $(T, \mu, \epsilon)$ equipped with an algebra-over-a-monad $(A, \rho)$ to the (augmented) simplicial object whose structure maps are given by the structure maps of the monad and its action on its algebra:
This simplicial object can be viewed as a resolution of $A$, in a sense explained below.
Let $\mathbf{E}$ be a category and let $(T, m: T T \to T, u: 1_{\mathbf{E}} \to T)$ be a monad on $\mathbf{E}$. We let $\mathbf{E}^T$ denote the category of $T$-algebras, and $U: \mathbf{E}^T \to \mathbf{E}$ the forgetful functor which is monadic, with left adjoint $F$.
Recall that the (augmented) simplex category $\Delta_a$, viz. the category consisting of finite ordinals^{1} 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 $\Delta_a$ is ordinal addition $[m]+[n] = [m+n]$. If $[n]$ is the $n$-element ordinal, then the terminal object $[1]$ carries a unique monoid structure and represents the “generic monoid”^{2}.
Similarly $\Delta_a^{op}$ is the walking comonoid. Since the comonad $F U$ on $\mathbf{E}^T$ can be regarded as a comonoid in the strict monoidal category of endofunctors $[\mathbf{E}^T, \mathbf{E}^T]$ (with endofunctor composition as monoidal product), there is a strong monoidal functor (or in fact a unique strict monoidal functor)
that takes the generic monoid $[1]$ to $F U$ and generally $[n]$ to $(F U)^{\circ n}$.
If furthermore $A$ is a $T$-algebra, there is an evaluation functor
and we have the following definition:
The bar construction $Bar_T(A)$ is the simplicial $T$-algebra given by the composite functor
The composite
will here be called the bar resolution of $A$.
In the notation of two-sided bar constructions, the bar construction would be written as $Bar_T(A) = B(F, T, A)$, and the bar resolution as $B(T, T, A)$.
To explain the sense in which $U Bar_T(A)$ is an acyclic resolution of (the constant simplicial object) $A$, we recall the fundamental décalage construction. Very simply, putting
the décalage functor on simplicial objects $C^{\Delta_a^{op}}$ (valued in a category $\mathbf{C}$) is the functor
Note that $D$ has a comonad structure (inherited from the comonoid structure on $[1]$ in $\Delta_a^{op}$, see also at décalage – comonad structure), and therefore $P$ also carries a comonad structure. Notice also that there is a comonad map $D \to [1]\circ !$ (where $[1]: 1 \to \Delta_a^{op}$ is left adjoint to $!: \Delta_a^{op} \to 1$ since $[1]$ is initial in $\Delta_a^{op}$), induced by the evident natural inclusion $[1]+i: [1]+[0] \to [1]+[m]$ in $\Delta_a$. This in turn induces a comonad map $P X \to {|X|}$ where ${|-|}$ is the composite (“discretization”):
The notation $P$ is chosen because décalage is essentially a kind of path space construction, i.e., in the case $\mathbf{C} = Set$ it is a simplicial sets analogue of a topological pullback
where $id: {|X|} \to X$ is the identity inclusion of the underlying set with the discrete topology. $P X$ is essentially a sum of spaces of based paths $(\alpha: (I, 0) \to (X, x_0)$ over all possible choices of basepoint $x_0$, fibered over $X$ by taking $\alpha$ to $\alpha(1)$. Each space of based paths is contractible and therefore $P X$ 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 $X: \Delta_a^{op} \to C$ is a $P$-coalgebra structure $X \to P X$.
Here a $P$-coalgebra structure on $X$ is the same as a right $D$-coalgebra (or $D$-comodule) structure, given by a simplicial map $h: X \to X \circ D$ satisfying evident equations. In more nuts-and-bolts terms, it consists of a series of maps $h_n: X([n]) \to X([n+1])$ satisfying suitable equations.
The map $h: X \to X D$ may be viewed as a homotopy. Again, turning to the topological analogue for intuition, the corresponding $h: X \to P X$ is a homotopy (or rather, the composite $X \to P X \to X^I$ can be turned into a homotopy $I \times X \to X$). The coalgebra structure $h: X \to P X$ has a retraction given by the counit $\varepsilon: P X \to X$, so $X$ becomes a retract of an acyclic space, hence acyclic itself.
Definition 2 gives an absolute notion of acyclicity, in the sense that if $X: \Delta_a^{op} \to \mathbf{C}$ carries an acyclic structure $h: X \to X D$ and $G: \mathbf{C} \to \mathbf{C}'$ is any functor, then $G X$ automatically carries an acyclic structure $G h: F X \to F X D$. (For example, $G X$ becomes acyclic in a standard model category sense under any functor $G: \mathbf{C} \to Set$.)
Returning now to the bar resolution $U Bar_T(A)$: there is a canonical natural isomorphism $T \circ U Bar_T \cong U Bar_T \circ D$ obtained as the following 2-cell pasting (where $U Bar_T$ abbreviates the top and bottom horizontal composites)
whence there is a homotopy
The map $h$ is an acyclic structure, def. 2, i.e., a right $D$-coalgebra structure.
We verify the coassociativity condition for the coaction $h: U Bar_T \to U Bar_T D$; the counit condition is checked along similar lines. The comultiplication of the comonad $F U$ is $\delta \coloneqq F u U$, and putting $\eta = u U: U \to U F U$ for a right $F U$-coaction, the coassociativity of $\eta$ follows from a naturality square
Apply $[id_{\mathbf{E}^T}, -]$ to this coassociativity square to get another coassociativity, this time for the comonad $K \coloneqq [id_{\mathbf{E}^T}, F U]$ on $[\mathbf{E}^T, \mathbf{E}^T]$ (with comultiplication denoted $\delta_K$) and coaction $H \coloneqq [id, \eta]: [id, U] \to [id, U] \circ K$. Thus there is an equalizing diagram
Because $Bar_T: \Delta_a^{op} \to [\mathbf{E}^T, \mathbf{E}^T]$ is a strong monoidal functor (see the left isomorphism in (1)), the squares in
commute serially, with the triangle commuting by definition of $h$. This completes the verification.
By Remark 1, it follows that $U Bar_T(A)$, obtained by applying evaluation at a $T$-algebra $A$, carries an acyclic structure as well. In this sense we may say that $U Bar_T(A)$ (which has $A$ as its augmented component in dimension $-1$) is an acyclic resolution of the constant simplicial $T$-algebra at $A$ that carries a $T$-algebra structure.
We now state and prove a universal property of the bar construction $Bar_T(A)$.
Let $(T, m: T T \to T, u: 1 \to T)$ be a monad on a category $\mathbf{E}$. A $T$-algebra resolution is a simplicial object $Y: \Delta_a^{op} \to \mathbf{E}^T$ together with an acyclic structure on $U Y: \Delta_a^{op} \to \mathbf{E}$. A morphism between $T$-algebra resolutions is a natural transformation $\phi: Y \to Y'$ such that $U\phi: U Y \to U Y'$ is a $P$-coalgebra map.
Let $AlgRes_T$ be the category of $T$-algebra resolutions. There is a forgetful functor
that takes an algebra resolution $Y$ to its augmentation component $Y[0]$.
The functor $\hom_{\mathbf{E}^T}(A, G-): AlgRes_T \to Set$ is represented by $Bar_T(A)$; i.e., $Bar_T(-): \mathbf{E}^T \to AlgRes_T$ is left adjoint to $G$.
The proof is distributed over two lemmas.
Given a $T$-algebra resolution $Y$ and a $T$-algebra map $f: A \to Y[0]$, there is at most one $T$-algebra resolution map $\phi: Bar_T(A) \to Y$ such that $\phi[0] = f$.
The $P$-coalgebra structure $h: U Bar_T(A) \to U Bar_T(A) \circ D$ is defined on components $U Bar_T(A)[n] = T^n A$ by $h[n] = u T^n(A): T^n(A) \to T^{n+1}(A)$. Thus in order that $U\phi$ be a $P$-coalgebra map, we must have that the diagram
commutes. Here $\phi[n]: T^n (A) \to Y[n]$ determines a unique $T$-algebra map $g: F T^n(A) \to Y[n+1]$ such that
since $F$ is left adjoint to $U$. Thus, starting with $\phi[0] = f$ as given, each algebra map $\phi[n]$ uniquely determines its successor $\phi[n+1] = g$.
The preceding proof does not show that the $\phi[n]$ fit together to form a map $\phi$ of simplicial $T$-algebras (i.e., to respect faces and degeneracies); it merely shows at most one such $T$-algebra resolution map is possible. But once we show that $\phi$ respects faces and degeneracies, the proof of Theorem 1 will be complete.
Given a $T$-algebra resolution $Y$ and an algebra map $f: X \to Y[0]$, there is at least one $T$-algebra resolution map $\phi: Bar_T(X) \to Y$ with $\phi[0] = f$.
It is enough to produce such a map $\phi: Bar_T(Y[0]) \to Y$ in the case $f = 1_{Y[0]}$, since the case for general $f: X \to Y[0]$ is then given by a composite
We will do something slightly more general. For any category $\mathbf{C}$, the endofunctor category $[\mathbf{C}^{\Delta_a^{op}}, \mathbf{C}^{\Delta_a^{op}}]$ has a comonoid object $P = - \circ D$, so that there is an induced strong monoidal functor
which, upon evaluating at an object $Y$ of $\mathbf{C}^{\Delta_a^{op}}$, gives a functor
with $B(Y, D, D)[n] = Y D^n$, so that $B(Y, D, D)$ is a double simplicial object. Taking $\mathbf{C} = \mathbf{E}^T$ and taking $Y$ to be a $T$-algebra resolution with acyclic structure $h: Y \to Y D$, we will produce a (double) simplicial map
where $\Phi[n]: T^n Y \to Y D^n$ is defined recursively as in the proof of Lemma 1, by setting $\Phi[0] = 1_Y$ and taking $\Phi[n+1]: T^{n+1} Y \to Y D^{n+1}$ the unique simplicial $T$-algebra map such that
commutes for all $n$. Once we verify the claim that $\Phi$ respects faces and degeneracies, the same will be true for $\phi[n] = \Phi[n][0]: (T^n Y)[0] = T^n(Y[0]) \to (Y D^n)[0] = Y[n]$, whence the proof will be complete by Remark 2.
The claim is proved by induction on $n$. Let $\epsilon: D \to 1_{\Delta_a^{op}}$ be the counit and $\delta: D \to D D$ be the comultiplication. We have face maps
for $j = 0$ to $n$, under the special convention that $m T^{-1}Y: T Y \to Y$ denotes the action $\alpha: T Y \to Y$. We also have degeneracy maps
for $j = 1$ to $n$. We proceed as follows.
To check preservation of face maps, we treat separately the cases where $j = 0$ and $j \geq 1$.
Since all the maps are algebra maps and $u T^n Y: T^n Y \to T^{n+1} Y$ exhibits $T^{n+1} Y$ as the free algebra on $T^n Y$, it suffices to check commutativity around the perimeter. (N.B.: the triangle commutes, even in the case where $n=0$ which we need to start the induction.) By definition of $\Phi[n+1]$, commutativity of the perimeter boils down to commutativity of
where the triangle commutes by one of the acyclic structure equations.
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 $\Phi[n]$. But the perimeter commutes by examining the diagram
(where the middle vertical arrow is $Y D^{j-1} \epsilon D^{n-j}$) using the inductive hypothesis in the bottom left parallelogram.
To check preservation of degeneracy maps, we treat separately the cases $j=1$ and $j \geq 2$.
is by appeal to a freeness argument where we just need to check commutativity of the perimeter (the special case $n=1$ being used to start the induction). But this boils down to commutativity of the diagram
where the bottom right quadrilateral commutes by one of the acyclic structure equations.
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
where the middle vertical arrow is $Y D^{j-2} \delta D^{n-j}$.
This completes the proof.
Let $A$ be a commutative associative algebras over some ring $k$. Write $A Mod$ for the category of connective chain complexes of modules over $A$.
For $N$ a right module, also $N \otimes_k A$ is canonically a module. This construction extends to a functor
The monoid-structure on $A$ makes this a monad in Cat: the monad product and unit are given by the product and unit in $A$.
For $N$ a module its right action $\rho :N \otimes A \to N$ makes the module an algebra over this monad.
The bar construction $\mathrm{B}(A,N)$ is then the simplicial module
Under the Moore complex functor of the Dold-Kan correspondence this is identified with a chain complex 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. Because the first authors denoted its elements using a notation involving vertical bars (Ginzburg)!!
This chain complex provides a resolution that computes the Tor
This gives the Hochschild homology of $A$. See there for more details.
See bar and cobar construction.
See (Fresse).
A general discussion of bar construction for monads is 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
N.B.: including the empty ordinal. ↩
If $X: \Delta_a^{op} \to \mathbf{C}$ is a simplicial object, then $X([n])$ is what is usually denoted $X_{n-1}$, the object of cells in dimension $n-1$. Note that $X([0]) = X_{-1}$ is the augmented component. The $n$ can be thought of as the number of vertices of a simplex of dimension $n-1$. We choose the index $n$ over the geometric dimension $n-1$ as it is more convenient for our purposes. ↩