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Aug
16
reviewed Close Does 'much' of what is known about groups carry over to groupoids?
Aug
16
reviewed Leave Open How to interpret this
Aug
16
comment Is the negation of the Gödel sentence always unprovable too?
On one level, it's Platonism – some people believe there is one "true" model of arithmetic, called $\mathbb{N}$. On another level, we're just thinking about provability in a bigger system. As I said before, don't read too much into the word "true".
Aug
16
comment Are there any known interesting F-(co)algebras where F isn't a set endofunctor?
No, for natural numbers you want the algebra for the endofunctor $X \mapsto 1 + X$. Similarly, lists are an algebra for the functor $X \mapsto A \times X$. The coalgebras for $X \mapsto A \times X$ are streams.
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
reviewed Edit suggested edit on Probability Density Function: Small question
Aug
15
revised Probability Density Function: Small question
Fixed formatting.
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
15
answered What is it about modern set theory that prevents us from defining the set of all sets which are not members of themselves?
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
revised Where else has Proposition B1.3.17 in the Elephant been proved?
edited tags
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.