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Aug
15
comment Totally ordered set with greater cardinality than the continuum
No axiom of choice is needed here: use the Hartogs number construction. See here, for example.
Aug
15
comment (Continued:) finiteness of étale morphisms
Hartshorne only defines ‘smooth of relative dimension $n$’ for morphisms of finite type. Being étale and of finite type is stable under base change, so the fibre over $x$ is étale and of finite type over $\operatorname{Spec} \kappa (x)$, where $\kappa (x)$ is the residue field at $x$ – and so the fibre must be a finite discrete set of points.
Aug
15
comment (Continued:) finiteness of étale morphisms
What's your definition of étale? If you only require an étale morphism to be locally of finite presentation, then an infinite disjoint union of copies of $\operatorname{Spec} k$ is étale over $\operatorname{Spec} k$, for obvious reasons.
Aug
14
comment Composition of derived functors and comparison between hypercohomology and sheaf cohomology
The theorem you cite is about derived categories and total derived functors. If you want to get anything concrete about cohomological derived functors you have to take a spectral sequence.
Aug
14
comment Which mappings are functors?
If you delete all non-invertible morphisms in the category you start with, then you get a functor. Otherwise what are you going to do with the morphisms?
Aug
14
comment Why bother proving that a class is a set?
If a class is a set, then there are a lot more things you can do to it: you can form its powerset, take subsets of it, etc. As for your unary "function" $f$ – if $f$ is a "small" function, then $f$ has a domain of definition, so the set of all sets closed under $f$ is going to be a set.
Aug
14
comment Which mappings are functors?
The automorphism group "assignment" is almost never a functor.
Aug
13
comment Questions about adjointness of quantifiers in first-order logic
$P \vdash Q$ in logic means $Q$ is derivable from $P$. $P \dashv \vdash Q$ is an abbreviation for "$P \vdash Q$ and $Q \vdash P$". A left adjoint preserves all coproducts, so if a functor doesn't preserve even one coproduct, it cannot be a left adjoint.
Aug
13
comment Where is the symmetric group hidden in the Yoneda lemma?
See this answer.
Aug
13
comment Terminology for the sheaf on Spec A
@ashpool Of course not. Let $k$ be any field, then $\operatorname{Spec} k[x]_{(x)}$ is the Sierpiński space – for any choice of $k$.
Aug
12
comment Where else has Proposition B1.3.17 in the Elephant been proved?
You may want to look at Streicher's notes on fibred categories, particular Chapters 12 and 13.
Aug
12
comment Where else has Proposition B1.3.17 in the Elephant been proved?
If $\mathbb{C}$ is an fibred/indexed category then $\mathbb{C}^I$ refers to the fibre of $\mathbb{C}$ over the object $I$. However, since $\pi_0 \mathcal{D}$ is just a set and $\mathcal{T}$ is an ordinary category, I suspect $\mathcal{T}^{\pi_0 \mathcal{D}}$ means the $\pi_0 \mathcal{D}$-fold product of $\mathcal{T}$ as an ordinary category.
Aug
11
comment Every poset is embedded into a meet-semilattice
@JD It isn't an answer – just a remark. The question deserves an answer phrased in purely order-theoretic terms.
Aug
11
comment Every poset is embedded into a meet-semilattice
This is a special case of the Yoneda embedding, which you can read about in any category theory textbook.
Aug
11
comment Nonzero elements of splitting field
@MihaHabič That's what I thought at first, and then I realised that there had to be some non-trivial cancellations happening.
Aug
11
comment Mordell equation implemented anywhere?
Since this amounts to finding integer points on elliptic curves, you might want to look at this MO question.
Aug
10
comment Is the Green-Tao primes theorem true or pseudo-true?
It is probably "pseudo-true". Many fundamental results in real analysis – such as the intermediate value theorem – are only "pseudo-true". But even if the Green–Tao proof isn't intuitionistically valid as-is, to answer the question properly one would have to either produce an intuitionistic counterexample to Green–Tao, or reprove the theorem using intuitionistic methods.
Aug
10
comment Is the Green-Tao primes theorem true or pseudo-true?
The second author writes here, "the argument was (necessarily) finitary in nature, but it was absolutely essential for us to be aware of the infinitary arguments and intuition that had been developed in ergodic theory, as we had to adapt such arguments to the finitary setting in order to conclude our proof, and it would have far less evident how to discover such arguments if we were always restricted to looking at finitary settings." (Not that I read the paper!)
Aug
9
comment What is the algebraic structure of functions with fixed points?
Congratulations: you have discovered the category of pointed sets/spaces/manifolds/etc.
Aug
9
comment Abstract nonsense proof of the cocompleteness of the category of groups
Here is a general nonsense proof that the category of algebras for a monad on $\textbf{Set}$ is cocomplete.