# Model structure for cooperads and for coalgebras

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?

Bibliography

[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

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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 »,
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.
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.