Symmetric Homotopy Theory for Operads

My first article (joint work with Bruno Vallette) has just appeared on arXiv! It deals with various constructions related to the homotopy theory of dg operads over arbitrary unital commutative rings: in particular, we define a higher cobar-bar adjunction and show that it provides us with a new cofibrant replacement functor, and introduce a notion of higher homotopy operads. Here, I would like to take the time to explain the motivation behind this work, sketch the approach we used, and talk about the meaning of the main results.

In case you can’t wait to read the actual article with all of its gory details, here it is. Of course, comments, criticism, and suggestions are always welcome!


The category of reduced augmented dg operads over a unital commutative ring $\mathbf{k}$ admits a model category structure [1,7,8] where the weak equivalences are the quasi-isomorphisms and the fibrations are the degree-wise epimorphisms. In this context, we will be interested in cofibrant replacements $\mathcal{Q}\overset{\sim}{\longrightarrow}\mathcal{P}$ for a given operad $\mathcal{P}.$ Algebras over a cofibrant operad $\mathcal{Q}$ satisfy a version of the Boardman–Vogt homotopy invariance property [1, Theorem 3.5]: given a homotopy equivalence of cofibrant-fibrant spaces $X,$ $Y,$ a $\mathcal{Q}$-algebra structure on either induces a homotopy equivalent $\mathcal{Q}$-algebra structure on the other.

One way to obtain such a cofibrant replacement for an operad is by means of the cobar-bar adjunction $$ \Omega : \textsf{conil dg Coop} \rightleftharpoons \textsf{aug dg Op} : \mathrm{B} , $$ as first introduced by Getzler–Jones [6]. The counit of this adjunction gives functorial cofibrant resolutions $\Omega\mathrm{B}\mathcal{P} \overset{\sim}{\longrightarrow} \mathcal{P}$ for operads $\mathcal{P}$ that are $\mathbb{S}$-cofibrant [2, §8.5], i.e. have cofibrant underlying $\mathbb{S}$-modules. Explicitly, this condition means that $\mathcal{P}(n)$ consist of projective $\mathbf{k}[\mathbb{S}_n]$-modules for all arities $n$ and in all degrees.

When $\mathbf{k}$ is a field of characteristic $0,$ the $\mathbb{S}$-cofibrancy condition is always satisfied: by Maschke’s theorem, the group algebras $\mathbf{k}[\mathbb{S}_n]$ are semisimple for all $n$ and hence all $\mathbf{k}[\mathbb{S}_n]$-modules are projective. When $\mathbf{k}$ is an arbitrary unital commutative ring, the cobar-bar adjunction still gives functorial resolutions $\Omega\mathrm{B}\mathcal{P} \overset{\sim}{\longrightarrow} \mathcal{P}$ for operads $\mathcal{P}$ whose underlying chain complexes consist of projective $\mathbf{k}$-modules [5, Theorem 2.1.15]. However, $\Omega\mathrm{B}\mathcal{P}$ is not cofibrant in general.

Motivation and approach

As we have seen in the previous section, the cobar-bar resolution resolves the operadic composition but not the symmetric group actions. The reason is that the symmetric group actions are not viewed as part of the operadic structure but live in the underlying $\mathbb{S}$-modules. In our article, we take a different perspective: we view operads as structure on underlying $\mathbb{N}$-modules, thereby viewing the symmetries of operations as part of the operadic data—on the same level as the operadic compositions. The goal is, of course, to find good cofibrant resolutions for operads over any unital commutative ring $\mathbf{k}.$

The way we do this is by adding an extra layer of abstraction: we consider an $\mathbb{N}$-colored operad $\mathcal{O}$ whose algebras are non-unital operads, a.k.a. pseudo-operads. We are not the first to consider this operad, see e.g. [3, §1.5.6], but we are the first to consider a presentation for it.

The presentation we use has the following generating operations: \begin{align} E(n;n) &:= \big\{ (␣)^\sigma \mid \sigma \in \mathbb{S}_n \setminus \{\textrm{id}_n\} \big\} , \cr E(n+k-1;n,k) &:= \big\{ (␣\circ_i␣) \mid 1 \leq i \leq n \big\} \cr &\qquad \cup \big\{ (␣\circ_j␣)^{(12)} \mid 1 \leq j \leq k \big\} , \end{align} with the evident $\mathbb{S}_2$-action mapping $(␣\circ_i␣) \in E(n+k-1;n,k)$ to $(␣\circ_i␣)^{(12)} \in E(n+k-1;k,n).$ These operations are required to satisfy three kinds of relations:

  1. the axioms of a right group action \begin{align} ((␣)^\sigma)^\tau &= (␣)^{\sigma\tau} , & \tau &\neq \sigma^{-1} , \cr ((␣)^\sigma)^{\sigma^{-1}} &= (␣) , \end{align}
  2. the usual associativity conditions for partial compositions \begin{align} ((␣\circ_j␣)\circ_i␣) &= ((␣\circ_i␣)\circ_{j+k-1}␣)^{(23)} , & i &< j , \cr (␣\circ_i(␣\circ_j␣)) &= ((␣\circ_i␣)\circ_{j+i-1}␣) , \end{align}
  3. and the compatibility relations between them \begin{align} (␣\circ_i(␣)^\sigma) &= (␣\circ_i␣)^{\sigma’} , \cr ((␣)^\sigma\circ_i␣) &= (␣\circ_{\sigma(i)}␣)^{\sigma’'} . \end{align}

Lemma. The algebras over the colored operad $\mathcal{O}$ are the non-unital operads.

Proof. An $\mathcal{O}$-algebra $\mathcal{P}$ consists of an $\mathbb{N}$-indexed collection of chain complexes—i.e. an $\mathbb{N}$-module—equipped with structure maps \begin{align} (␣)^\sigma &\colon \mathcal{P}(n) \longrightarrow \mathcal{P}(n) , &&\sigma \in \mathbb{S}_n \setminus \{\textrm{id}_n\} , \cr (␣\circ_i␣) &\colon \mathcal{P}(n) \otimes \mathcal{P}(k) \longrightarrow \mathcal{P}(n+k-1) , && 1 \leq i \leq n . \end{align} The relations are the usual ones for right $\mathbb{S}_n$-actions, operadic partial compositions, and compatibility between them.

Homotopy symmetric operads

Let’s have a closer look at the presentation for $\mathcal{O}$ given in the previous section. Note that the associativity and compatibility relations are of quadratic type, while the group action axioms are quadratic-linear resp. quadratic-constant. In particular, the canonical choice of augmentation $\mathcal{O}\to\mathrm{I}$ mapping $\mathrm{id}_n=(␣)\in\mathcal{O}(n;n)$ to $\mathrm{id}_n\in\mathrm{I}(n;n)$ and all generators to $0,$ is in fact not a morphism of operads, and thus not actually an augmentation. Because of this, we cannot apply the usual cobar-bar construction to $\mathcal{O}.$

Luckily for us, Hirsh–Millès [9] extended the cobar-bar adjunction to semi-augmented dg operads and conilpotent curved cooperads, $$ \Omega : \textsf{conil curved Coop} \rightleftharpoons \textsf{sdg Op} : \mathrm{B} . $$ A semi-augmentation for an operad is just like an augmentation, except that a semi-augmentation is only required to be a morphism of (non-dg) $\mathbb{S}$-modules. In particular, a semi-augmentation is not assumed to be a morphism of operads. On the cooperad side, the result is a derivation that doesn’t necessarily square to $0.$ This defect is controlled by a curvature term. A notion of curved twisting morphisms is introduced and shown to be represented by the curved cobar and bar constructions on the left resp. on the right, i.e. $$ \begin{matrix} \operatorname{Hom}_\textsf{sdg Op}(\Omega\mathcal{C},\mathcal{P}) &\cong& \operatorname{Tw}(\mathcal{C},\mathcal{P}) &\cong& \operatorname{Hom}_\textsf{conil curved Coop} (\mathcal{C},\mathrm{B}\mathcal{P}) . \end{matrix} $$

In addition, Hirsh–Millès developed a curved Koszul duality theory: given a semi-augmented dg operad $\mathcal{P}$ with a quadratic-linear-constant presentation, a Koszul dual (conilpotent, curved) cooperad $\mathcal{P}^¡$ is constructed. As usual, the cooperad $\mathcal{P}$ is called Koszul if a certain (curved) twisting morphism $\kappa\colon\mathcal{P}^¡\to\mathcal{P}$ is Koszul, i.e. corresponds to a quasi-isomorphism $\Omega\mathcal{P}^¡\to\mathcal{P}.$

Theorem 1. The colored operad $\mathcal{O}$ is curved Koszul.


The statement can be shown e.g. using the usual rewriting approach [11, §8.3] with rewriting rules given by rewriting the left-hand-side of the relations as the right-hand-side. This approach, however, is a bit tedious as it requires checking confluency of many diagrams.

Another way to prove the theorem—and the one we use in our article—is to make the observation that the operad $\mathcal{O}$ can be written as $\mathcal{O} = \mathbf{k}[\mathbb{S}] \circ_\Lambda \mathcal{O}_\textrm{ns}$ using a distributive law [11, §8.6] $$ \Lambda\colon \mathcal{O}_\textrm{ns} \circ \mathbf{k}[\mathbb{S}] \longrightarrow \mathbf{k}[\mathbb{S}] \circ \mathcal{O}_\textrm{ns} . $$ Here, we denote by $\mathbf{k}[\mathbb{S}]$ the colored suboperad of $\mathcal{O}$ generated by the unary generators, and by $\mathcal{O}_\textrm{ns}$ the colored suboperad generated by the binary generators. Explicitly, $\mathbf{k}[\mathbb{S}]$ consists of the group algebras concentrated in arity $1,$ i.e. $(\mathbf{k}[\mathbb{S}])(n;n) = \mathbf{k}[\mathbb{S}_n],$ while $\mathcal{O}_\textrm{ns}$ is the colored operad of non-symmetric non-unital operads. With this approach, we can use the fact that the operad $\mathcal{O}_\textrm{ns}$ is Koszul auto-dual [10, Theorem 4.3] and that $\mathbf{k}[\mathbb{S}]$ is Koszul. As a result, we only need to check confluency for mixed rewriting diagrams.

The cofibrant resolution $\mathcal{O}_\infty:=\Omega\mathcal{O}^¡\longrightarrow\mathcal{O}$ of the operad $\mathcal{O}$ allows us to define a notion of operads up to homotopy with the desired homotopy properties over any unital commutative ring.

Definition. A higher homotopy operad is an $\mathcal{O}_\infty$-algebra $\mathcal{P}$ or, equivalently, a twisting morphism $\mathcal{O}^¡\to\operatorname{End}_\mathcal{P}.$

In our article, we make this definition explicit in terms of structure maps and relations using the curved twisting morphism bifunctor. We compare our higher homotopy operads with other notions of operads up to homotopy, in particular Brinkmeier’s lax operads [4] and van der Laan’s strongly homotopy operads [10].

Cofibrant resolutions for operads

Let $\mathcal{C}$ be a curved dg cooperad and $\mathcal{P}$ a semi-augmented dg operad. Any curved twisting morphism $\alpha\colon\mathcal{C}\to\mathcal{P}$ gives rise to a cobar-bar adjunction [9] between the categories of (co)algebras $$ \Omega_\alpha : \textsf{conil $\mathcal{C}$-Coalg} \rightleftharpoons \textsf{$\mathcal{P}$-Alg} : \mathrm{B}_\alpha . $$ In our case, we apply this general construction to the canonical Koszul twisting morphism $\kappa\colon\mathcal{O}^¡\to\mathcal{O}$. This leads to an adjunction $$ \Omega_\kappa : \textsf{conil $\mathcal{O}^¡$-Coalg} \rightleftharpoons \textsf{$\mathcal{O}$-algebras} : \mathrm{B}_\kappa . $$ We have already seen that $\mathcal{O}$-algebras are precisely non-unital dg operads. The $\mathcal{O}^¡$-coalgebras are a new kind of cooperads and differ from the normal cooperads in that they come with certain symmetric group coactions in addition to the partial decompositions instead of the underlying symmetric group actions normal cooperads have. We call such an $\mathcal{O}^¡$-coalgebra a higher dg cooperad and make this definition explicit in terms of structure maps and relations in our article. From now on, we use the notation $\widetilde{\Omega}$ resp. $\widetilde{\mathrm{B}}$ for $\Omega_\kappa$ resp. $\mathrm{B}_\kappa$ and call $$ \widetilde{\Omega} : \textsf{conil higher dg Coop} \rightleftharpoons \textsf{nu dg Op} : \widetilde{\mathrm{B}} $$ the higher cobar-bar adjunction.

Recall that the category of non-unital dg operads is equivalent to the category auf augmented dg operads. Any augmented dg operad $\mathcal{P}$ splits into $\mathcal{P} = \mathrm{I} \oplus \bar{\mathcal{P}}$, and the augmentation ideal $\bar{\mathcal{P}} = \ker{\epsilon}$ is a non-unital dg operad. Conversely, one can formally adjoin a unit to any non-unital dg operad to obtain an augmented dg operad.

By [9, §5.2], the higher cobar-bar adjunction provides us with cofibrant resolutions $\widetilde{\Omega}\widetilde{\mathrm{B}}\bar{\mathcal{P}} \overset{\sim}{\longrightarrow} \bar{\mathcal{P}},$ however the result is shown only over a field $\mathbf{k}$ of characteristic $0.$ We give an explicit proof of the analogous statement for the respective augmentations for any unital commutative ring $\mathbf{k}.$

Theorem 2. The augmentation of the higher cobar-bar counit, $$ \mathrm{I}\oplus\widetilde{\Omega}\widetilde{\mathrm{B}}\bar{\mathcal{P}} \overset{\sim}{\longrightarrow} \mathcal{P} , $$ gives a functorial resolution of dg operads $\mathcal{P}$ which is cofibrant when the underlying chain complexes are non-negatively graded and consist of projective $\mathbf{k}$-modules.

A careful analysis of this resolution shows that it is, in fact, isomorphic to an existing construction: arity-wise tensoring with the Barratt-Eccles operad $\mathcal{E}$ to obtain an $\mathbb{S}$-cofibrant resolution, followed by the usual cobar-bar resolution. We construct the isomorphism explicitly.

Theorem 3. There exists a natural isomorphism of augmented dg operads $$ \mathrm{I} \oplus \widetilde{\Omega}\widetilde{\mathrm{B}}\bar{\mathcal{P}} \cong \Omega\mathrm{B}(\mathcal{P}\otimes\mathcal{E}) $$ between the augmentation of the higher cobar-bar resolution of $\bar{\mathcal{P}}$ and the cobar-bar resolution of the arity-wise tensor product of the operad $\mathcal{P}$ with the Barratt–Eccles operad $\mathcal{E}.$

Applications and further research

Consider the operad $\mathcal{P}=\mathrm{Com}.$ Using the results above, we know that $$ \mathrm{I}\oplus\widetilde{\Omega}\widetilde{\mathrm{B}}\bar{\mathrm{Com}} \cong \Omega\mathrm{B}(\mathcal{E}) \overset{\sim}{\longrightarrow} \mathrm{Com} $$ defines a cofibrant resolution for the operad $\mathrm{Com}$ of commutative algebras over any unital commutative ring $\mathbf{k}.$ In other words, $\Omega\mathrm{B}(\mathcal{E})$ is a cofibrant $\mathrm{E}_\infty$ operad. Now this was already known. Nonetheless, our results do show a certain universality of the construction and hence provide a more conceptual understanding.

Similarly, for the operad $\mathrm{Lie}$ we obtain a cofibrant resolution $$ \mathrm{I}\oplus\widetilde{\Omega}\widetilde{\mathrm{B}}\bar{\mathrm{Lie}} \cong \Omega\mathrm{B}(\mathrm{Lie}\otimes\mathcal{E}) \overset{\sim}{\longrightarrow} \mathrm{Lie} $$ and hence algebras over $\Omega\mathrm{B}(\mathrm{Lie}\otimes\mathcal{E})$ deserve the name homotopy Lie algebras over any unital commutative ring $\mathbf{k}.$

As for the usual cobar-bar adjunction $\Omega\dashv\mathrm{B},$ we would like to have a higher Koszul duality theory in the context of our higher cobar-bar adjunction $\widetilde{\Omega}\dashv\widetilde{\mathrm{B}}.$ The goal, of course, is to replace $\widetilde{\mathrm{B}}\mathcal{P}$ by a higher Koszul dual, i.e. a certain higher dg subcooperad $\mathcal{P}^\diamond\subset\widetilde{\mathrm{B}}\mathcal{P}$, in order to obtain smaller resolutions. A candidate for such a theory is the Koszul duality for algebras over an operad as developed by Millès [12, §3]. In its current version, however, it applies only to algebras with a monogene presentation. Neither the operad $\mathrm{Com}$ nor the operad $\mathrm{Lie}$ admit a (homogeneous) monogene presentation as algebras over the operad $\mathcal{O}$.


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