I am studying the homotopy theory of (algebraic) operads and I came up with several questions I am unable to answer to. I would like to stress that I don't have applications in mind, I just would like to understand the "state of art".

Initial assumption. In what follows, every operad is assumed to be reduced.

Let me recall a few well-known facts:

  1. fix a closed symmetric monoidal model category $(\mathcal E, \otimes, I)$, which is moreover cofibrantly generated. It is shown in Spitzweck's thesis [1] that the collection of operads in $\mathcal E$ has always a J-semi-model structure; moreover, this becomes a model structure whenever $\mathcal E$ has a cocommutative Hopf interval in the sense of [2];
  2. Following [2], we say that an operad $P$ is admissible if $P\textrm{-} \mathrm{Alg}$ obtains a model structure which is transferred from $\mathcal E$. Again in [1] it is shown that if $(\mathcal E, \otimes, I)$ satisfies the monoid axiom of [2], any cofibrant operad (w.r.t. the semi model structure) is admissible;
  3. more generally, if $P$ is an operad endowed with a map $P \to P \otimes Q$ (where this is the Hadamard product) and the category $\mathcal E$ has an interval endowed with a $Q$-coalgebra structure, then $P$ is admissible. As corollary, we obtain that if there exists a cocommutative coassociative interval, then every operad is admissible. This is shown in [2].

Point 3. seems really interesting to me: it relates algebras and coalgebras in a sort of pairing. Let me be more precise: if we knew that there is a map of operads $P \to P \otimes Q$ and we knew moreover that the category $Q \textrm{-} \mathrm{coAlg}$ of $Q$-coalgebras has a model structure where weak equivalences and cofibrations are defined via the forgetful functor, then the condition stated in 3. would hold trivially. However I cannot hope to use the transfer principle, because the adjunction (forgetful,cofree) for coalgebras goes in the wrong direction. My first question is therefore:

Question 1. Is there in the literature a notion of "coadmissible operad", i.e. an operad $P$ such that $P\textrm{-}\mathrm{coAlg}$ has a model structure such that the forgetful functor to $\mathcal E$ preserves weak equivalences and cofibrations?

Changing somehow approach and flavour, I began to study B. Vallette's Homotopy Theory of Homotopy Algebras [4]. I learned that when $\mathcal E = \mathrm{dgVect}_k$, if $P$ is Koszul then conilpotent $P^¡$-coalgebras do have a model structure satisfying Question 1 (this makes sense because coalgebras over a cooperad are simply coalgebras over the linear dual operad). This model structure is obtained via the bar-cobar adjunction induced by the obvious twisting morphism $\kappa \colon P^¡ \to P$, but it is not a simple application of the transfer principle: once again, the direction of the adjoint functors is the wrong one and one needs to work a lot with spectral sequences (using conilpotency) to get the result. Therefore I am led to the following question:

Question 2. Is there in the literature a notion of "admissible twisting morphism"? Let us say that this could mean the following: a twisting morphism $\alpha \colon C \to P$ from a cooperad to an operad is said to be admissible if the category of conilpotent $C$ coalgebras have a model structure where cofibrations are degreewise monomorphisms and a map $f$ is a weak equivalence iff $\Omega_\alpha(f)$ is a weak equivalence. Here $\Omega_\alpha$ is the bar construction, part of the bar-cobar adjunction $\Omega_\alpha \colon \text{conil } C\textrm{-}\mathrm{coAlg} \leftrightarrows P \textrm{-}\mathrm{Alg} \colon B_\alpha$.

Remark. I am not too strict on the above definition of admissible twisting morphism. My question is: "has someone studied something looking similar?".

Finally, the last question is:

Question 3. Is there a treatment of cooperads similar to the one given in [2]? Is there a general principle comparable with "cofibrant operads are admissible"? If not, can someone explain me the obstructions?


[1] M. Spitzweck, Operads, Algebras and Modules in General Model Categories, avaiable at http://arxiv.org/abs/math/0101102

[2] C. Berger, I. Moerdijk, Axiomatic homotopy thoery for operads.

[3] S. Schwede, B. Shipley, Algebras and modules in monoidal model categories.

[4] B. Vallette, Homotopy theory of homotopy algebras, available at http://math.unice.fr/~brunov/HomotopyTheory.pdf


2 Answers 2


Question 1. I do not know precisely. But here are some references that address the question of model category structures on cooperads and coalgebras over a cooperads.

  • Aubry—Chataur « Cooperads and coalgebras as closed model categories »,
  • Hess—Shipley « The homotopy theory of coalgebras over a comonad »,
  • Getzler—Goerss « A model category structure for differential graded coalgebras » (available on Paul Goerss Home Page).

Question 2. I would like to stress that the model category structure that I need on conilpotent $P^¡$-coalgebra is different from the one(s) developed in the aforementioned references. Here the class of weak equivalences is not the class of quasi-isomorphisms. Actually, one gets the "quasi-isomorphism » model category structure by Bousfield localisation from that one. This phenomenon is similar to what is going on for simplicial sets. Again, I have never seen any paper where such a notion of admissible twisting morphism is studied in depth. (Of course, feel free to write one!) So far, one can say that the Koszul morphisms $\kappa : P^{¡} \to P$ coming from the Koszul duality theory are admissible.

Question 3. If one has in mind a model category structure similar to the one written in [4] but on the level of cooperads (instead of coalgebras), then all the cooperads are cofibrant …


Joey Hirsh and I have given an answer to question 2. We assume our operads and cooperads are $\Sigma$-split outside characteristic zero for the usual technical reasons and additionally require a weight grading on the cooperad so that we can use Bruno's results. Then any twisting morphism is admissible in the sense of the question.

These structures are functorial in both cooperad and operad and promote the bar and cobar functors to Quillen adjoints, which (in characteristic zero, at least) are Quillen equivalences if and only if the twisting morphism is Koszul.


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