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See [LV] Loday–Vallette, Algebraic Operads, Section 8.6 for all the definitions.

Let $\mathtt{P} = \mathbb{F}(V)/(R)$ and $\mathtt{Q} = \mathbb{F}(W)/(S)$ be two quadratic operads presented by generators and relations, and let $$\lambda : W \circ_{(1)} V \to V \circ_{(1)} W$$ be a rewriting rule. Let $\mathtt{P} \vee_\lambda \mathtt{Q}$ be the resulting operad, a quotient of $\mathbb{F}(V \oplus W)$.

By Theorem 8.6.5 in [LV], it is sufficient for the projection $\mathtt{P} \circ \mathtt{Q} \to \mathtt{P} \vee_\lambda \mathtt{Q}$ to be injective when restricted to operations of total weight $3$ in generating operations for the rewriting rule to induce a distributive law (basically, the underlying symmetric sequence of $\mathtt{P} \vee_\lambda \mathtt{Q}$ is isomorphic to $\mathtt{P} \circ \mathtt{Q}$).

Some elements of $W \circ_{(1)} V \circ_{(1)} V$, namely those of the form $$q(1,\dots,1,p,1,\dots,1,p',1\dots,1),$$ $$\text{and } q(1, \dots, p(1, \dots, p', \dots, 1), \dots, 1)$$ are roughly speaking "critical monomials" and can be rewritten in two ways, inducing relations in $\mathtt{P} \vee_\lambda \mathtt{Q}$. If $p$ is injective when restricted to weight $3$, then all of these are confluent, i.e. the relations already come from $R$ and $S$.

Question. If all these "critical monomials" are confluent, is $p$ injective in weight $3$? (And hence $\lambda$ induces a distributive law on $\mathtt{P} \circ \mathtt{Q}$ by Theorem 8.6.5 above?)

My intuition, attempts at proofs, and trying on some simple examples, all say yes. However I've not been able to come up with a full proof, nor with a counterexample. The case I'm most interested in has an infinite number of generating operations, so brute force seems suboptimal.

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  • $\begingroup$ Let's see if I understand. Your question is whether confluence $\Rightarrow$ injectivity in weight $3$, right? And then you use Theorem 8.6.5 to deduce that injectivity in weight $2$ implies that the projection is an isomorphism. $\endgroup$
    – Fernando Muro
    Commented Jun 16, 2016 at 13:45
  • $\begingroup$ @Fernando Yes, exactly. $\endgroup$
    – Najib Idrissi
    Commented Jun 16, 2016 at 13:51
  • $\begingroup$ Najib, I've been dealing these days with this sections' results, and it worries me that Theorem 8.6.5 uses the filtration in Theorem 8.6.4 which, unlike claimed, is not compatible with the differential. Theorem 8.6.4 can be fixed. I haven't check Theorem 8.6.5 though. $\endgroup$
    – Fernando Muro
    Commented Jun 16, 2016 at 13:56
  • $\begingroup$ @Fernando Oh, I see. Thank you. I hope I won't need the result too badly... $\endgroup$
    – Najib Idrissi
    Commented Jun 16, 2016 at 15:17
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    $\begingroup$ There is one more confluence to check here. There are also elements of the form $q\circ_i(p\circ_j p')$, which can be rewritten in two different ways, using the rewriting rule twice right away, or using the relations of the operad $P$ and then the rewriting rule twice. I cannot see how you take care of these by just considering the elements you are considering. $\endgroup$
    – Vladimir Dotsenko
    Commented Nov 4, 2016 at 18:10

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