Timeline for Monoids with infinite products
Current License: CC BY-SA 2.5
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 22, 2013 at 1:09 | vote | accept | David Spivak | ||
| Jun 25, 2011 at 12:41 | comment | added | Gerald Edgar | Your example of $\mathbb N$ extended to $\mathbb N \cup \{\infty\}$ is a good one. But, as we see from the answers below, you cannot even expect $\mathbb Z$ to have such an extension. | |
| Jun 25, 2011 at 0:28 | answer | added | Benjamin Steinberg | timeline score: 4 | |
| Sep 13, 2010 at 11:14 | answer | added | Martin Brandenburg | timeline score: 0 | |
| Sep 9, 2010 at 0:31 | answer | added | Theo Johnson-Freyd | timeline score: 0 | |
| Sep 8, 2010 at 23:37 | answer | added | user6976 | timeline score: 2 | |
| Sep 8, 2010 at 23:25 | answer | added | Todd Trimble | timeline score: 5 | |
| Sep 8, 2010 at 21:06 | comment | added | David Spivak | As for Qiaochu Yuan's question -- yes, this is what I mean by "identities can be thrown out." It also follows from the "concatenation of sequences" axiom. As for Peter's question -- good point. Just to make sure the idea is clear at least, the colimit of a finite sequence ($\kappa=n$ for some finite n) will include a canonical map from $0\in [n]$ and this map will be the product. So in the finite case, the ideas coincide. Colimits also take care of the other properties (think "final subcategory.") But that's not to say that I can show that colimits are really what I want here. | |
| Sep 8, 2010 at 21:01 | comment | added | David Spivak | Sorry, I've been absent -- you all have good points. I'm not exactly sure what I want here; perhaps that is clear. I'm looking for the right notion. As for Mark's question -- you're right to ask about inverses: I was sloppy. I think the question of whether the addition of infinite products to a commutative monoid leaves it commutative is a matter of choice, at least in the "add products freely" functor. Under the "add a catch-all" it will remain commutative. | |
| Sep 8, 2010 at 20:14 | comment | added | Peter LeFanu Lumsdaine | I don't see how your "another way to phrase this" is the same thing... how do the axioms in the first paragraph force the infinite products to be colimits? | |
| Sep 8, 2010 at 19:50 | comment | added | Qiaochu Yuan | How about requiring that the product of (m_1, m_2, ...) is equal to the product of (1, m_1, m_2, ...)? | |
| Sep 8, 2010 at 19:06 | comment | added | user6976 | So far the question does not make much sense (for me). What do you mean by "inverses"? Consider the monoid given by one defining relation $ab=1$ (the polycyclic monoid; note that $ba\not = 1$ in this monoid). What do you want to do with it? If your original monoid is commutative, do you expect the result to be commutative too? | |
| Sep 8, 2010 at 18:50 | history | edited | David Spivak | CC BY-SA 2.5 |
changed title to reflect internal terminology
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| Sep 8, 2010 at 18:29 | history | asked | David Spivak | CC BY-SA 2.5 |