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Sep
22
comment Ring homomorphism
This is almost completely irrelevant: the question is asking about ring homomorphisms, not group homomorphisms.
Sep
22
comment The existence of ends of functors.
Cantor's theorem says $2^\kappa > \kappa$, and because $\kappa$ is strongly inaccessible, it must lie outside $V_\kappa$.
Sep
21
awarded  Custodian
Sep
21
comment Matrices of forms seen as sections of a vector bundle
Yes, basically. Or you can think of it as $\Omega^p \otimes E \otimes E^*$ if you prefer.
Sep
21
accepted Pullbacks of categories
Sep
21
comment The existence of ends of functors.
Universes are very simple things: $U$ is a universe in ZF if and only if $U = V_\kappa$ for a strongly inaccessible cardinal $\kappa$. With respect to a fixed universe $U$, a set is "small" if and only if it is in $U$. Nothing mysterious at all!
Sep
21
comment Infinitely generated modules
Or even more boring, $\mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z} \oplus \cdots$ over $\mathbb{Z}$.
Sep
21
comment What are $\partial/\partial f^j$ in Jost's definition of the differential mapping?
The components of $f$ are with respect to some coordinate system $y^i$, and the author is abusing notation by writing $\frac{\partial}{\partial f^i}$ for $\frac{\partial}{\partial y^i}$. But it's perfectly understandable: this way, it looks like the pushforward is nothing more than the chain rule.
Sep
21
comment The existence of ends of functors.
It isn't always. See the second example.
Sep
20
comment The existence of ends of functors.
That's the easiest way to formalise things. It's somewhat troublesome to define things like $\textrm{Nat}(F, G)$ in NBG or MK class–set theory when the categories involved are large.
Sep
20
answered The existence of ends of functors.
Sep
20
comment The existence of ends of functors.
This has nothing to do with set theory. If $\mathcal{X}$ is only known to have small limits then you simply can't talk about $\prod_c S(c, c)$ when $c$ varies over a non-small set (i.e. a proper class).
Sep
20
comment The existence of ends of functors.
Sometimes it exists, sometimes it doesn't. The usual construction only works when $\mathcal{X}$ has enough limits.
Sep
20
comment Why does the definition of homotopy cartesian involve factorisations
Consider the insertion of two different points into a contractible space. The strict pullback of these points is the empty space, but the homotopy pullback is non-empty and contractible, as one would expect. So the strict pullback doesn't respect weak equivalences in any meaningful sense.
Sep
20
comment Homomorphism of free modules $A^m\to A^n$
But $I$ is not free, no?
Sep
19
comment Some questions about rings
The direct sum of infinitely many rings is not even a ring.
Sep
19
comment Axiom of choice question
Of course, one could add a choice operator to the logical system, in which case one really can infer $\varphi (c)$ from $\exists x . \varphi (x)$, where $c = \tau_x \varphi (x)$...
Sep
19
comment Why use ZF over NFU?
Sure, you can define that map. But there's no internal bijection $X \to \mathscr{P}_1(X)$, so for category-theoretic purposes it doesn't satisfy the requirements.
Sep
19
comment Need a FOL derivation validated. Unclear on universal generalization restriction.
That looks highly suspect to me.
Sep
19
answered showing logical argument is valid or invalid