3
$\begingroup$

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!

Improve this question
$\endgroup$
1
  • 1
    $\begingroup$ Instead of reading the answers, 1) Understand the case $C=(\mathrm{Set},\times,1)$, 2) Guess the general formula for monoidal categories (with coproducts which distribute over the tensor product) 3) Prove it. There is no reason to look this up in a book, it's a good exercise. $\endgroup$
    – Martin Brandenburg
    Commented Feb 7, 2012 at 10:54

2 Answers 2

Reset to default
7
$\begingroup$

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.

Improve this answer
$\endgroup$
1
  • $\begingroup$ Thanks a lot. Forgot to look in MacClane, which is usually my first stop! $\endgroup$
    – Jonathan Beardsley
    Commented Feb 7, 2012 at 0:19
2
$\begingroup$

The 2nd edition of Mac Lane's Categories for the Working Mathematician has a description/construction of the free monoid on a given object in a monoidal category satisfying some side conditions, see Section 7.3. I am not sure off the top of my head if those conditions are met for every symmetric closed monoidal category, but they certainly hold for Set, K-Mod, Ban${}_1$ and other familiar examples.

(I don't think you can get free monoidal objects in Ban, but I have never written down exact details, and suspect that's not really an example of direct interest to you.)

Improve this answer
$\endgroup$
4
  • $\begingroup$ Thanks a lot. What is Ban? $\endgroup$
    – Jonathan Beardsley
    Commented Feb 7, 2012 at 0:17
  • $\begingroup$ Ban denotes the category of Banach spaces and continuous linear maps. Ban_1 means the category of Banach spaces and continuous linear maps of norm bounded above by 1. There is some discussion of these categories in the nLab article on Banach spaces: ncatlab.org/nlab/show/Banach+space $\endgroup$
    – Todd Trimble
    Commented Feb 7, 2012 at 0:36
  • $\begingroup$ Sorry Jon, was in a hurry and forgot to clarify; didn't mean to be needlessly esoteric. Todd's comment has it exactly right. $\endgroup$
    – Yemon Choi
    Commented Feb 7, 2012 at 0:42
  • $\begingroup$ Oh no problem. Thanks to all parties involved. Very interesting though. $\endgroup$
    – Jonathan Beardsley
    Commented Feb 7, 2012 at 2:13

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .