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14h
comment Category-theoretic characterization of zero-dimensional spaces
In particular, "clopen sets separate points" can be encoded categorically very simply by saying "every monomorphism $2\to X$ has a left inverse", where $2$ is the coproduct of two copies of the terminal object. But I'm not sure if this is the sort of thing you're looking for...
15h
comment Category-theoretic characterization of zero-dimensional spaces
What would you consider "similar"? The category CHaus is rigid (every self-equivalence is naturally isomorphic to the identity), so in some sense every object in CHaus can be characterized categorically.
17h
comment Cyclic faithfully flat modules
This is trivially impossible for commutative rings (if $R/I$ is faithfully flat, tensor it with $0\to I\to R\to R/I\to 0$ to find that $I=0$).
1d
revised Lattice Flatness Measure
edited tags
2d
comment An exercise in the Kaplansky's book
More conceptually: if $i$ is an idempotent, any idempotent in $R/(i)$ lifts to an idempotent in $R$ (this is obvious when you think of idempotents as decompositions of the ring as a product). Any idempotent in $R/I$ is already idempotent in some $R/I_0$ where $I_0\subseteq I$ is generated by finitely many idempotents. Since the ideal generated by finitely many idempotents is just the ideal generated by their join, it follows that any idempotent in $R/I$ lifts to an idempotent in $R$.
Aug
28
revised Does $\mathbb Z \times \mathbb Z$ mod the obvious $\mathbb Z$ action have more structure than just a set?
edited tags
Aug
28
awarded  Tag Editor
Aug
28
wiki created lattice-theory description
Aug
28
wiki created lattice-theory excerpt
Aug
27
comment A Linear Order from AP Calculus
@DmitryV: That's not what I was talking about (I was also thinking like a category theorist and not caring about that). If $f/g$ oscillates between approaching $0$ and approaching $\infty$, then $f\not\leq g$ and $g\not\leq f$. Actually, if $f$ and $g$ don't have to be positive, you could just have an unbounded set where $f$ is zero and $g$ is not and an unbounded set where $g$ is zero and $f$ is not.
Aug
27
comment A Linear Order from AP Calculus
This is not a total order; $f$ and $g$ can oscillate between $f\gg g$ and $g\gg f$.
Aug
25
comment Which algebraic relations are possible between algebraic conjugates?
Clearly the orbit of $\alpha$ when you iterate $f$ must be finite, which gives a simpler argument for your last two examples.
Aug
25
comment On the Riesz representation theorem II
This is Theorem 3.10 in Rudin's Functional Analysis.
Aug
25
answered On the Riesz representation theorem II
Aug
24
answered On the Riesz representation theorem
Aug
24
comment On the Riesz representation theorem
Do you want that limit to just hold pointwise, or do you want something stronger? If you just want it pointwise, the Riesz representation theorem has nothing to do with this--you can find such a net for any $\Phi$ (because you can find a $\phi$ that works for any finite set of $\psi$s). On the other hand, if you restrict to sequences, $\Phi$ does have to be bounded if $V$ is complete, but this is not obvious (it follows from the Banach-Steinhaus theorem).
Aug
20
answered Hedetniemi's conjecture for graphs with countable chromatic number
Aug
19
revised Can the Kan-Thurston theorem be turned into some kind of equivalence between groups and spaces?
made title less generic
Aug
19
reviewed Leave Open Can the Kan-Thurston theorem be turned into some kind of equivalence between groups and spaces?
Aug
18
comment Reference for (co)limit-preserving functor $X\mapsto R^X$
@OwenBiesel: I don't know any reference for this exact argument (you can cite this very MO post if you want such an exact reference...), but many of the ideas involved are discussed in Peter Johnstone's book Stone spaces (in particular, some constructions with my functor $G$ for $R=\mathbb{Z}$ (but not its left adjoint) are discussed in section V.2 and my last paragraph is discussed in section VI.2.3)