I posted a question a week ago on math.stackexchange. As is sometimes the case, I got no answers. Considering that the question is about a research article, I hope that it might be relevant for MathOverflow.

Here is the original question:

I’m having trouble with the proof of Lemma 2.9 in “Cohomology of Monoids in Monoidal Categories” by Baues, Jibladze, and Tonks, and I was wondering if someone could clarify a detail. I’ll try to summarize the context of the lemma.

## Context

Let $(Bbb A,circ,I)$ be an monoidal category where $Bbb A$ is abelian: in particular, $circ$ is not necessarily additive in both arguments. Suppose that $circ$ is left distributive, i.e. the natural transformation

$$(X_1circ Y)oplus(X_2circ Y)rightarrow (X_1oplus X_2)circ Y$$

is an isomorphism. For example, $Bbb A$ could be the category of linear operads (this is a motivating example of the article). Given an endofunctor $F$ of $Bbb A$, we define its *cross-effect*

$$F(A|B):=ker(F(Aoplus B)rightarrow F(A)oplus F(B)).$$

The *additivization* of $F$ is then the functor $F^text{add}$ defined by

$$F^text{add}(A):=text{coker}left(F(A|A)rightarrow F(Aoplus A)xrightarrow{F(+)}F(A)right).$$

The idea is that $F^text{add}$ is the additive part of $F$.

Let $(M,mu,eta)$ be an internal monoid in $Bbb A$, and let $L_0$ be the endofunctor of $Bbb A$ defined by $L_0(A)=Mcirc(Moplus A)$. Let $L:=L_0^text{add}$ be the additivization of $L_0$. (In the case of operads, represented as planar trees, I see $L(A)$ as the space of trees whose nodes are all labeled by elements of $M$ except for one leaf, which is labeled by an element of $A$.)

Suppose now that $Bbb A$ is *right compatible with cokernels*, i.e. that

for each $AinBbb A$, the additive functor $Acirc-:Bbb ArightarrowBbb A$ given by $Bmapsto Acirc B$ preserves cokernels.

Then, in the proof of Lemma 2.9, the authors claim the following:

By the assumption that $Bbb A$ is right compatible with cokernels it follows that $L(L(X))$ is the additivisation of $L_0(L_0(X))$ in $X$ (…).

## Remarks

If anyone could **provide an explanation of the last claim**, I would be very grateful. However, my inability to understand how to show this might be related to two other issues I have:

1) Elsewhere in the literature, cross-effects are only defined when $F$ is *reduced*, i.e. $F(0)=0$ (e.g. here, section 2). But we can always reduce a functor by taking the cokernel of $F(0)rightarrow F(X)$, so I don’t think it’s much of a problem.

2) In the first quote, the authors state that $Acirc -$ is additive, which is quite the opposite of the initial hypothesis that $circ$ be *left* distributive, and not necessarily *right* distributive. How to resolve this apparent conflict?