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53m
comment Hard-to-synthesize but easy-to-prove results
One is reminded of a joke: "According to the Nobel Prize-winning physicist Richard Feynman, mathematicians designate any theorem as "trivial" once a proof has been obtained--no matter how difficult the theorem was to prove in the first place. There are therefore exactly two types of true mathematical propositions: trivial ones, and those which have not yet been proven."
18h
comment Invertibility of Units/Counits
That's on p. 92 in the second edition. I suppose you have the first edition. Anyway, the claim follows from the fact that $F \eta$ is invertible if and only if $G \epsilon$ is invertible. See here.
20h
comment Invertibility of Units/Counits
What? The word "either" doesn't seem to appear on the page at all.
1d
comment Not every over-under-category is cocomplete
The category in question is a disjoint union of model categories. Perhaps that suffices for whatever purposes intended.
1d
comment Do Natural transformations make 'God given' precise?
Many such "god-given" morphisms are natural and vice versa. But one can always concoct counterexamples.
1d
comment Rational Points, classical versus modern notion
You might be able to do this for $\mathbb{Q}$, but only because it's the prime field. I doubt you would be able to identify, say, the $\mathbb{R}$-rational points without first choosing an embedding into affine space.
2d
comment Restriction of sheaf via inclusion induces isomorphism on stalks
Restriction does indeed commute with colimits, because it's a left adjoint.
Sep
19
comment Is a pushout of monics still monic?
@AmitaiYuval It is not true in $\mathbf{CRing}$. A counterexample is contained in my answer here.
Sep
19
comment Is a pushout of monics still monic?
@skysurf3000 I think for your specific question the answer is yes, but you just have to prove it by hand.
Sep
18
comment Where do divisors on curves come from?
I suspect they are called "divisors" by analogy with the theory of number fields.
Sep
18
comment Why is axiom of choice needed? (Equivalent conditions for Noetherian)
The proof as quoted uses dependent choice.
Sep
18
comment Fibred product of schemes
There's no real difference if you ignore the underlying topological space and instead focus on the $A$-points for arbitrary rings $A$.
Sep
17
comment Cambridge Maths Tripos Papers
I'm sure you'll find them in the university library.
Sep
17
comment 2-category in HoTT: chapter 9 from the HoTT book
That's almost surely a joke exercise, along the same lines as Lang's famous "Take any book on homological algebra, and prove all the theorems without looking at the proofs given in that book."
Sep
17
comment 2-category in HoTT: chapter 9 from the HoTT book
Why would you want to define a (pre)category inside a 2-category? The point of a 2-category is that its objects are already category-like.
Sep
17
comment Formal schemes vs formal power series
It's something like the difference between a module for $k [[x]]$ as an ordinary ring and a module for $k [[x]]$ as a topological ring.
Sep
17
comment Continuous maps vs. open maps
Oops, forgot that I could close unilaterally. If someone strongly disagrees, feel free to reopen.
Sep
17
comment $H_0(X) \cong \tilde{H}_0(X) \oplus \mathbb{Z}$ isomorphism in algebraic topology
For the isomorphism, you need to use the fact that $\mathbb{Z}$ is projective or free.
Sep
14
comment Notation for a functor between comma categories
$(\Delta \downarrow \phi)$ is not unreasonable.
Sep
12
comment example of two non-isomorphic fields which embed inside each other
Answer: Yes, you can. Hint: Try something with infinite transcendence degree over its prime field...