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Dec
12
comment Does the $O(n)$ bundle of a manifold depend on the metric?
I'm confused. The group of orthogonal transformations of $T_x M$ is indeed isomorphic to $\mathrm{O}(n)$, but not in any canonical way: indeed, choosing an isomorphism amounts to choosing an orthonormal basis for $T_x M$. So although we can construct a group bundle over $M$ with typical fibre $\mathrm{O}(n)$, I don't see how we can get a canonical $\mathrm{O}(n)$-action on it.
Dec
12
comment Different Negations of Self-referential Propositions
I think the point is that a meaningless sentence is neither true nor false, c.f. "Colourless green ideas sleep furiously." That is not to say self-referential propositions are necessarily meaningless, but it does require some care to make them meaningful...
Dec
12
comment About covering maps and sections!
Well, what if $E = X \amalg X$?
Dec
12
comment Union of categories and $\ell$-adic local systems
If I had to guess, I would say that it is obtained by taking the inverse limit of all the categories of $K$-local systems, as $K$ varies over the finite extensions of $\mathbb{Q}_\ell$. But that is just a guess.
Dec
12
comment Automorphism of an elementary extension of a structure that moves an undefinable element
Write $h_i : \mathfrak{A}_i \to \mathfrak{B}$ for the colimit homomorphisms. The automorphism $g$ is the unique homomorphism such that $g \circ h_0 = h_2 \circ g_1 \circ f_0$, $g \circ h_1 = h_2 \circ g_1$, $g \circ h_2 = h_4 \circ g_3 \circ f_2$, $g \circ h_3 = h_4 \circ g_3$, etc.; its inverse is described similarly. (Note that $g$ does not have to be self-inverse!)
Dec
11
comment Is the halting of a program that checks for duplicates in an infinite multiset decidable?
If the set $\Sigma$ is presented as an infinite string of zeros and ones (i.e. as its characteristic function) then you would have to read the entire list before being able to determine whether it is empty – obviously, this will take an infinite amount of time. If we cut down to recursive sets, so that $\Sigma$ can be presented by the program that determines membership, we still wouldn't be able to effectively decide whether that program ever says that something is in $\Sigma$ or not.
Dec
11
comment Finite quotient ring of $\mathbb Z[X]$
Or, in other words, it is a preorder.
Dec
11
comment understanding covariant derivative (connexion)
Yes, you're right. If it were so easy to define a connection then the space of connections would naturally be a vector space, rather than just an affine space!
Dec
11
revised understanding covariant derivative (connexion)
deleted 103 characters in body
Dec
11
comment Determine Consistency Of System Specifications
I think you are supposed to take all three statements as hypotheses and see whether or not an inconsistency can be derived from them.
Dec
11
answered How to characterize categories which their only isomorphisms are identities?
Dec
11
comment How to characterize categories which their only isomorphisms are identities?
These are called gaunt categories.
Dec
11
comment How to characterize categories which their only isomorphisms are identities?
That's not correct. In a skeletal category there can be non-trivial automorphisms. What is true is that all isomorphisms are automorphisms.
Dec
11
comment Verifying that a function is a morphism by checking a generating set
This question only makes sense for concrete categories, and very concrete ones at that: these generating sets are really just sets and not objects in the same category.
Dec
11
comment Is the halting of a program that checks for duplicates in an infinite multiset decidable?
Yes, but even then there are still problems, as you pointed out in your answer. I was trying to suggest that the naive interpretation of the question is uninteresting: as soon as you allow inputs of infinite length, then any program that checks for duplicates in the input would have to run forever, and so it fails to halt in a very trivial way.
Dec
11
comment Inductive vs projective limit of sequence of split surjections
I don't see why not. Instead of taking the whole of $\mathbb{Z}^{\times \mathbb{N}}$, we could take the submodule generated by $\mathbb{Z}^{\oplus \mathbb{N}}$ (which is countable) and the sequence $(1, 1, 1, \ldots)$; in other words, this is the space of eventually-constant sequences.
Dec
11
comment What is the difference between necessary and sufficient conditions?
Those are the definitions of necessary and of sufficient.
Dec
11
answered Automorphism of an elementary extension of a structure that moves an undefinable element
Dec
11
comment Is Hom$(G,-)$ left exact if morphisms are required to be continuous?
$\textrm{Hom}(-, G)$ is very rarely right exact, but again, if it fails to be left exact, you have defined exact sequence incorrectly.
Dec
11
awarded  algebraic-geometry