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4h
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...
5h
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.
21h
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?")
22h
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.
23h
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!
Jul
29
comment Do direct limits (filtered colimits) commute with pullbacks, in C*-algebras?
To find a counterexample, we could start by concentrating on commutative C*-algebras. The full subcategory Comm-C*-Alg of C*-Alg is closed under limits and colimits, so it doesn't matter which category we think of the (co)limits as happening in. Now Comm-C*-Alg is dual to the category CptHff of compact Hausdorff spaces, so to find a counterexample, it's enough to show that inverse lims (cofiltered lims) don't commute with pushouts in CptHff. Taking inverse lims does preserve coproducts and epis in CptHff, so those two simple kinds of pushout won't help us... we need something a bit cleverer.
Jul
28
comment When did coordinate plane “as we know it” come into play?
Imagine if someone nowadays wrote a paper whose title started "An instance of the Excellence of Modern ALGEBRA". You'd immediately dismiss them as a crank. Have we lost something? Or do we just prefer our titles not to sound like Bill and Ted?
Jul
21
revised Is this graph of reciprocal power means always convex?
added 381 characters in body
Jul
21
comment Is this graph of reciprocal power means always convex?
Thanks very much, Dirk and Robert. I agree with Dirk: it's still a puzzle as to why it's so nearly true. E.g. in this particular example, the non-convexity is extremely subtle - I just plotted the graph and couldn't detect it by eye.