How general is $TX \otimes X \simeq X \otimes TX$?

Question

Let $C$ be a monoidal category, $X \in C$ an object, and $TX$ the free monoid object on $X$, assuming it exists. How often do we have an isomorphism $TX \otimes X = X \otimes TX$? is there a canonical map, or maybe a canonical (co)span, between them in general? or anything that characterizes a canonical isomorphism when it exists?

For more details: In the special case where $C$ has colimits and they are preserved by $\otimes$ in each variables, we can write $$ TX = \bigoplus_{n \geqslant 0} X^{\otimes n} $$ so that $$X \otimes TX \simeq \bigoplus_{n \geqslant 0} X^{\otimes (n+1)} \simeq TX \otimes X.$$

But I am interested in cases where this formula for $TX$ doesn't hold, and it seems this formula is true well beyond that case. For example, if I look at the example of the composition product of collection, then for a collection $C$, $TC$ is the free operad on a collection, so is described in terms of $C$-labelled trees, and both $C \otimes TC$ and $TC \otimes C$ can be understood as $C$-labeled tree of height at least $1$, and it feels very similar to the previous isomorphism…. I've also checked many cases where $X$ is an endofunctor and $TX$ is the free monad on $X$, not clear it works in general, but it seemed to work on all the examples I managed to understand.

I feel like the special case where $TX$ is computed by some transfinite process is more likely to be true (as the only strategy I can think of is to use an explicit description of $TX$, though I have to admit at this point I find it is not so clear how to get it from the more general formula for $TX$). So I guess I have three related questions:

1. Do we always have $TX \otimes X \simeq X \otimes TX$, or is there a counterxample where $TX$ exists but there is no such isomorphism?

2. Does this hold when $C$ has all colimits and $\otimes$ preserves $\kappa$-filtered colimits, or maybe even filtered colimits, in each variable, so that we can use a transfinite process to compute $TX$?

3. In the general case, is there some sort of characterization of the kind of isomorphism I mentioned above? Like a comparison map in one direction that might not be an isomorphism in general, but specializes to the isomorphisms above in nice cases?

Answer 1

I believe that, contrary to your claim, this is not true for what you call collections, that is for the monoidal category where the monoids are operads. Indeed, if $C$ is supported at arity 2 and is just the trivial $S_2$-module in that arity, $TC\circ C$ has nothing in arity 3 (since $\circ$ is the full substitution, so you have arity 2 when substituting in the unit element, and arity $2n$ when substituting into something of arity $n$), while $C\circ TC$ has such elements (use the unit of $TC$ in one slot, and the basis element of $C\subset TC$ in the other one).