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Dec
22
answered Reduction under the exponential transpose
Dec
21
comment What about split cones?
Huh, apparently I was mistaken. See the fourth bullet point here.
Dec
21
comment What about split cones?
There is a notion of absolute limit, but I don't think these are always split in quite the sense you suggest.
Dec
21
comment A sheaf of cumulative hierarchies
Ick. This looks like a terrifying hybrid of the cumulative hierarchy of set theory and sheaf theory. It is indeed based on forcing techniques (more precisely, boolean-valued models) from set theory, but you won't find it in Mac Lane and Moerdijk's book.
Dec
20
comment Elliptic curve terminology confusion
@Matt In the vast majority of cases, nothing is recoverable from just knowing the set! But you would have a more than decent chance of recovering $E$ if you had $E (K)$ as an algebraic variety (in the classical sense) and $K$ an algebraically closed field...
Dec
20
comment Is $(gf)(X) = g ( f(X))$ in a category?
If $\mathcal{C}$ has pullbacks, then images are functorial, in the sense that pulling back along a morphism $f : A \to B$ produces a functor $f^{-1} : \textrm{Sub}(B) \to \textrm{Sub}(A)$, and then the left adjoint $\exists_f : \textrm{Sub}(A) \to \textrm{Sub}(B)$ (if it exists) gives images. Of course, left adjoints are unique up to unique isomorphism if they exist, so $\exists_g \exists_f \cong \exists_{g \circ f}$.
Dec
20
comment Axiomatic characterization of the rational numbers
This is a second-order characterisation, though.
Dec
20
comment Is $(gf)(X) = g ( f(X))$ in a category?
Images are in general not useful without some structure on the ambient category. If $\mathcal{C}$ has pullbacks, then images behave sufficiently well that they compose as you expect.
Dec
19
comment Difference between a stalk of a sheaf and a fiber of a vector bundle
That is precisely the sense I mean. A pullback is a limit.
Dec
19
comment Why surjectivity stable under base change?
You need the following fact: if $K \to L$ and $K \to L'$ are a pair of field extensions, then there is a field $M$ and a pair of field homomorphisms $L \to M, L' \to M$ making the obvious diagram commute.
Dec
19
revised Difference between a stalk of a sheaf and a fiber of a vector bundle
added 453 characters in body
Dec
19
answered Difference between a stalk of a sheaf and a fiber of a vector bundle
Dec
19
comment Examples of fields of characteristic 1
Most likely the downvote was for the suggestion that $\{ 0 \}$ is a field...
Dec
19
comment Clarifications about the definition of algebraic systems and algebraic structures
Hmmm. I think by ‘algebraic structure’ most people mean a set with operations but no relations.
Dec
19
comment Clarifications about the definition of algebraic systems and algebraic structures
Your definitions are rather vague. Can you give exact quotations?
Dec
19
accepted Automorphism extension property of Galois extensions
Dec
19
comment What guarantees that non-geometric definition of trigonometry is actually the same as the geometric definition?
The standard definition of $\pi$ in analysis is "the first positive zero of $\sin$", so it's automatic there as well.
Dec
18
comment What is wrong with my proof? Every extension is separable? (of course not)
What if $p'$ is the zero polynomial?
Dec
18
comment Definition of differential of map (in algebraic geometry)
One doesn't usually have a cotangent bundle so much as a cotangent sheaf, in which case it is not appropriate to use fibre product notation for pullback.
Dec
18
comment Hartshorne Exercise II. 3.19 (b)
If you want a hint instead of a complete solution, then say so in your question. Otherwise this is a perfectly valid answer.