It is shown in Moriya (Multiplicative formality of operads and Sinha’s spectral sequence for long knots, 2.1) that there exists a left proper model category structure on non-symmetric operads over $k$-chain complexes $\textrm{Ch}(k)$, where $k$ is a field. He gives a direct proof of the theorem, and I am not sure whether the hypothesis of $k$ field is used somewhere.

I am trying to understand if this holds for $k= \mathbb{Z}$ too. In Muro (Homotopy Theory of non-symmetric operads), a model structure is shown to exist in much greater generality, and it is shown that if $\textrm{Ch}(k)$ is combinatorial, then so is the model category on $\textrm{Ch}(k)$ non-symmetric operads. However, left properness is not treated, and I can't figure out in an easy way what the generating cofibrations should b (to prove myself lef-properness).

EDIT: all chain complexes here are meant to be non negatively graded.

  • $\begingroup$ arxiv.org/abs/1304.6641 $\endgroup$ Jul 26, 2023 at 17:59
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    $\begingroup$ These papers have an erratum which doesn’t affect the part on operads. I copy here link in case you need anything about algebras: arxiv.org/abs/1507.06644 $\endgroup$ Jul 26, 2023 at 18:00
  • $\begingroup$ The arxiv links contain references to the published versions. $\endgroup$ Jul 26, 2023 at 18:02
  • $\begingroup$ @FernandoMuro: The link should probably be posted as an answer, to ensure the question does not keep showing up on the home page. $\endgroup$ Jul 26, 2023 at 18:12
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    $\begingroup$ The answer should be longer for one needs conditions on the ring for operads to be left proper, in general it’s just relatively left proper. $\endgroup$ Jul 26, 2023 at 18:18

1 Answer 1


Let $k$ be a commutative ring and let $\mathrm{Ch}(k)$ be the category of non-negatively graded chain complexes of $k$-modules. We endow it with the projective model structure. Weak equivalences are quasi-isomorphisms and fibrations are maps which are surjective in positive degrees. As a consequence of Theorem 1.11 in…

Muro, Fernando. “Homotopy Theory of Non-Symmetric Operads, II: Change of Base Category and Left Properness.” Algebraic & Geometric Topology 14, no. 1 (2014): 229–81. https://doi.org/10.2140/agt.2014.14.229.

… we obtain the following corollary:

Corollary: If all objects in $\mathrm{Ch}(k)$ are cofibrant then the category of non-symmetric operads in $\mathrm{Ch}(k)$, with weak equivalences and fibrations defined arity-wise, is left proper.

This sufficient condition holds for $k$ a field, but not for $\mathbb{Z}$.

It is easy to see that the category of non-symmetric operads can’t be left proper if there exists a $k$ module $M$ and $i>0$ such that $\mathrm{Tor}^k_i(M,M)\neq 0$ (this rules out $\mathbb{Z}$). Indeed, let $f\colon P\to M$ be a projective resolution. We can regard $f$ as a morphism of operads concentrated in arity $0$ (the arity $1$ part should be the identity in $k$ because of the operatic unit, but the rest is $0$). Then, if we freely add an arity $2$ generator to the source and target of $f$, we obtain a direct sum of tensor powers of $f$, including $f\otimes f\colon P\otimes P\to M\otimes M$, which is a quasi-isomorphism iff $\mathrm{Tor}^k_i(M,M)=0$ for all $i>0$. Notice that this doesn’t exclude commutative Von Neumann regular rings which are not fields.

In general, by the aforementioned theorem, we have a more general result, very close to left properness, which is sometimes enough for applications.

Theorem: A push-out of a weak equivalence of non-symmetric operads with underlying cofibrant complexes along a cofibration of operads is always a weak equivalence.

Notice that, over $\mathbb{Z}$, a complex is cofibrant if it consists of free abelian groups. This is the case for operads arising as singular or cellular chains on some geometric object.

The previously cited paper is a continuation of…

Muro, Fernando. “Homotopy Theory of Nonsymmetric Operads.” Algebraic & Geometric Topology 11, no. 3 (2011): 1541–99. https://doi.org/10.2140/agt.2011.11.1541.

These papers deal with both operads and algebras. I made a mistake in the algebra part of the first paper, which spread to the second paper. This doesn’t affect the operad part, which is what you’re interested in, but just in case let me copy the reference to the erratum:

Muro, Fernando. “Correction to the Articles ‘Homotopy Theory of Nonsymmetric Operads’, I-II.” Algebraic & Geometric Topology 17, no. 6 (2017): 2–3852. https://doi.org/10.2140/agt.2017.17.3837.

Hope this helps!

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    $\begingroup$ Thank you very much Fernando! I am flattered to get an answer by you in person! Indeed, thanks to the theorem you cited, I managed to show what I wanted, in combination with some nlab proposition on base-change. That is, in the hypothesis of thm 1.11, if V has in addition cofibrant tensor unit, then the base change $\textrm{Ass}/Op(V) \to B/Op(V)$ is a Quillen equivalence for $B \to \textrm{Ass}$ being a cofibrant resolution. This is true for $V=Ch(k)$ no matter which $k$, since the unit is always cofibrant. No left-properness needed! $\endgroup$ Jul 27, 2023 at 7:53
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    $\begingroup$ @AndreaMarino I’m happy to help! $\endgroup$ Jul 27, 2023 at 10:08
  • $\begingroup$ Hi Fernando, what about symmetric operads? Is there a convenient model structure, maybe induced by the forgetful functor (that possesses an adjoint)? I am particularly interested in comparing slice categories $\textrm{NonSymOp}_{O/}$ and $\textrm{SymOp}_{U(O)/}$, where $U(O)(n) = O(n) \times \Sigma_n$ is the associated free symmetric operad. $\endgroup$ Sep 29, 2023 at 16:40

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