I have a complete and co-complete monoidal category in which the tensor product commutes to filtered colimit in each variables, and I would like to understand the "free monoid" construction in this category.
One knows, for example by the work of E.Dubuc, that this always exists, but his constructions has some problems in the $\omega$ case and always require an induction process up to a uncountable regular cardinal, even in the $\omega$ case.
I was wondering if one can come up with a construction that only require a countable transfinite construction.
There is this paper of S.Lack which gives a nice answer in the case where the tensor product commutes to reflexive co-equalizer and countable composition. His construction does not seem to be what I'm looking for (in the situation I'm concerned reflexive co-equalizer are not conserved in general and are very messy to compute).
But in the introduction of his paper (linked above) S.Lack mention that in the case of a tensor product which preserve filtered co-limit one can use a formula of the form:
$I + X \otimes (I+X \otimes (I+ X \otimes (...$
He does not give much explanation for this, he seem to imply that this comes from the paper of Kelly on transfinite construction, but I can't find it in the paper, and I haven't been able to figure out the details by myself.
Does someone know what is this construction he is talking about ? Is there other "countable" construction of the free monoid that you know of ?
