There appear to be questions perhaps tangentially related to this that have been asked already. If so a reference and a close would be heartily appreciated.


Give some category $\mathcal{C}$ with the modifiers listed in the title of the question, and an object $A$ in that category, how does one define the "free monoid" on that object? Is this related to being an algebra over some monad? 


Thanks!



Free Monoids in Closed Symmetric Monoidal Categories
Very simply, if a closed symmetric monoidal category has countable coproducts, and if monoid means monoid with respect to the monoidal product, then the free monoid on an object $A$ can be constructed as the "geometric series" 


$$F(A) = \sum\_{n \geq 0} A^{\otimes n}.$$ 


The key fact needed to prove this is that $\otimes$ distributes over countable coproducts, and this is guaranteed by the closedness (indeed, $X \otimes -$ preserves arbitrary colimits). The proof that this is correct must be in thousands of places; see for instance Categories for the Working Mathematician (2nd edition), page 172. 


