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answered What is the opposite category of $Set$?
Jan
28
comment Relations commuting with logical equivalence.
Cross-posted to MO: mathoverflow.net/questions/195069/…
Jan
26
revised What is a Horn Clause?
added 78 characters in body
Jan
22
revised Do hom-sets really live in the category Set?
link to a cross-post
Jan
22
comment Do hom-sets really live in the category Set?
Now posted at MO as well: mathoverflow.net/questions/194551/…. An answer there was given by Simon Henry.
Jan
17
comment When is it possible to interpret composition as a natural transformation?
(1) I take it you mean a cartesian closed category. (2) Internal composition is an example of an extranatural transformation: ncatlab.org/nlab/show/extranatural+transformation It's dinatural in $Y$.
Jan
15
comment Construction of Yoneda extension
Yeah, $\theta(c)$ means the same as $\theta_c$ -- one sees both notations in the literature. The arrow you're asking about, if we rewrite the domain as $\sum_c \sum_{x \in P(c)} F(c)$, is the one whose restriction to the $x^{th}$ copy of $F(c)$ is given by the inclusion of the copy of $F(c)$ indexed by $y = \theta_c(x) \in Q(c)$ in $\sum_c \sum_{y \in Q(c)} F(c)$. Hope that helps.
Jan
15
comment Construction of Yoneda extension
It's also called a copower. See tac.mta.ca/tac/reprints/articles/10/tr10.pdf, page 54 of 143 (page 48 of the book), where the dot notation is used (it's commonly used in category theory). I'm not sure which you meant by the second equation, but essentially I'm using the co-Yoneda lemma: ncatlab.org/nlab/show/co-Yoneda+lemma . The advantage of this formulation (over a category of elements description) is that it generalizes straightforwardly to the enriched context, whereas the category of elements description does not.
Jan
13
answered Construction of Yoneda extension
Jan
12
comment Barycentric subdivision
This could be useful: tarunchitra.com/papers/6510/hw8.pdf (see page 5).
Jan
11
comment Can you equip every vector space with a Hilbert space structure?
So to answer one of the questions directly: this implies that some (real or complex) vector spaces do not admit a Hilbert space structure. Such as any whose dimension is the limit $\lambda$ of cardinals $\alpha_n$ defined by $\alpha_0 = \aleph_0$ and $\alpha_{n+1} = 2^{\alpha_n}$ (this $\lambda$ has countable cofinality and hence can't be of the form $\lambda = \kappa^{\aleph_0}$, else $\lambda = \lambda^{\aleph_0}$ which would contradict König's theorem, as mentioned in a comment above).
Jan
10
answered Can you equip every vector space with a Hilbert space structure?
Jan
10
comment Can you equip every vector space with a Hilbert space structure?
@AdamP.Goucher I don't think that's necessarily true. If $\kappa$ is a cardinal greater than the continuum but of countable cofinality, then $\kappa^{\aleph_0} > \kappa > 2^{\aleph_0}$. See en.wikipedia.org/wiki/…
Dec
23
comment Hilbert Spaces: Tensor Product
Really it is not a good idea to modify your question so as to render obsolete any answer(s) thoughtfully provided. Better is to ask a new question.
Dec
19
awarded  Constituent
Dec
19
answered Question about milnor's proof of hairy ball theorem
Dec
11
comment Paul Erdos's Two-Line Functional Analysis Proof
Sure would be nice to know what the result was (and the contents of Erdős's proof). Any update on that?
Dec
11
comment
Two things I would want in a moderator are transparency and an even temper. The exchange I read here doesn't give me much confidence in either for this candidate.
Dec
11
comment Conjecture about $A f(x) = f(g(x)) + f(h(x))$
I'm not following this question. Suppose $f$ is a constant function, say $f = 1$ everywhere, and $A = 1.5$. The expression $f(g(x)) + f(h(x))$ is $2$ everywhere and $f(g(x)) - f(h(x))$ is $0$ everywhere. So what gives?
Dec
11
comment Are two bimodules isomomorphic as left and right modules also isomorphic as bimodules?
This isn't really a research level question, but you might consider a situation where $R$ is an $R$-bimodule in the standard way, vs. a situation where $R$ is a left module in the standard way but a right module in a twisted way, twisted by an involution on the ring. Think of a familiar ring or field with an involution, and you should be on your way.