bio | website | maths.ed.ac.uk/~tl |
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visits | member for | 5 years, 1 month |
seen | 1 hour ago | |
stats | profile views | 6,844 |
Nov 20 |
awarded | Nice Answer |
Nov 9 |
comment |
What measurable quantity can constrain the number of odors human can discriminate?
I'm dismayed at the votes to close and Stefan's critique. Yoav carefully explained the issues in such a way that a mathematician with no knowledge of this biological system could make a meaningful contribution. If we close this question, we might as well put a banner on the front page saying "applied math questions not welcome here". |
Oct 24 |
answered | Topological characterization of injective metric spaces |
Oct 23 |
comment |
Characterize the category of rings
@Paul Yes! That was why I assumed that I wasn't really answering Chris's question. As you say, it's kind of a tautological statement. |
Oct 23 |
comment |
Characterize the category of rings
OK: the category of rings together with its forgetful functor to $\mathbf{Set}$ is terminal among all categories $C$ equipped with a functor $U: C \to \mathbf{Set}$ and a ring structure on the object $U \in [C, \mathbf{Set}]$! That's probably not the kind of thing you want, though... |
Oct 23 |
comment |
Characterize the category of rings
Do you definitely want to characterize the category of rings, rather than the category of rings together with its forgetful functor to Set? Not that I know how to do either, but I think the two questions are significantly different in character. |
Oct 22 |
comment |
Finitely co/continuous monad induced by an operad
Another little observation: if $T$ is a finitary monad on Set (such as the one induced by an operad) and the forgetful functor from $T$-algebras to Set preserves colimits, then $T$ is of the form $M \times -$ for some monoid $M$. That doesn't quite fit your situation, but begins to suggest that there aren't any nontrivial examples. (It's Lemma 1.2 of Peter Johnstone's paper "When is a variety a topos?") |
Oct 22 |
comment |
Finitely co/continuous monad induced by an operad
Still commenting rather than answering because I haven't got anything nontrivial to say... but if you view a monoid $M$ as an operad (with no operations of arity $\neq 1$) then the resulting monad is $M \otimes -$, which preserves colimits if your monoidal category is closed. But that's all I can think of. |
Oct 22 |
comment |
Finitely co/continuous monad induced by an operad
Preservation of terminal objects seems like a tall order, at least when the monoidal category is cartesian. If the operad is called $P$ then this would give $\sum_n P_n = \hat{T}(1) = 1$. But maybe you're not interested in the cartesian case. If not, could you give a hint as to what kind of example you are interested in? |
Oct 17 |
awarded | Yearling |
Oct 16 |
comment |
About the hypothesis of Zorn's lemma
Can one of the voters-to-close explain why they're voting to close? |
Oct 16 |
comment |
How short can we state the Axiom of Choice?
@FrodeBjørdal: this is set-theoretic! Although Paul used the word "topos", $X$, $Y$ and $Z$ are sets for your purposes. It might not be the kind of set theory you want to think about, but it's set theory all the same. |
Oct 13 |
awarded | Nice Answer |
Oct 12 |
comment |
When taking the fixed points commutes with taking the orbits
This result, including Will Sawin's extension of it in his answer below, is now written up: arxiv.org/abs/1409.7860 |
Oct 2 |
comment |
Why do roots of polynomials tend to have absolute value close to 1?
That's a great question. Have you taken a look at the pictures created by Dan Christensen (jdc.math.uwo.ca/roots) and the stuff John Baez has written about it (math.ucr.edu/home/baez/roots)? See also johncarlosbaez.wordpress.com/2011/12/11/the-beauty-of-roots and Jordan Ellenberg here: quomodocumque.wordpress.com/2010/01/09/… |
Sep 30 |
awarded | Explainer |
Sep 29 |
comment |
Why have mathematicians used differential equations to model nature instead of difference equations
@RichardHardy Welcome to MO! Maybe your point is that pure mathematics does not actually touch "nature"; it's not about the external world...? That I would agree with. Still, I disagree with what you write: there are plenty of mathematicians, both in math departments and in other departments such as biology or engineering, who spend much of their time in a quest to accurately model nature. |
Sep 5 |
comment |
Is every compact topological ring a profinite ring?
For discussion of this result, see here: golem.ph.utexas.edu/category/2014/08/… . In particular, Todd Trimble gives a very nice, short proof:golem.ph.utexas.edu/category/2014/08/… . I don't know whether it's the same as Ribes and Zalesski's. |
Sep 2 |
awarded | Necromancer |
Jul 29 |
comment |
Do direct limits (filtered colimits) commute with pullbacks, in C*-algebras?
@ToddTrimble: oh, oops. I didn't think that through. Never mind! |