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Oct
18
comment Other ways of proving that the set of all countable ordinals is uncountable
Not Russell's paradox so much as Burali-Forti's paradox, but they're closely related.
Oct
18
comment Projective module over a ring
$R^\kappa$ for an infinite cardinal $\kappa$ is not free in general. You mean $R^{\oplus \kappa}$.
Oct
18
comment The image of the spec functor under a restriction
The dual of the category of abelian groups can be embedded in the category of locally compact abelian groups, by Pontryagin duality.
Oct
18
comment The image of the spec functor under a restriction
What's a prime ideal in a non-unital ring?
Oct
18
comment Is there a quicker, nicer way to show that the union of compact sets is not necessarily compact?
Why not take $S_n = \{ n \}$? Then $S$ is just an infinite discrete set, which is obviously not compact.
Oct
17
revised How to show that $\Delta[n]$ isn't Kan fibrant…?
edited tags
Oct
17
answered How to show that $\Delta[n]$ isn't Kan fibrant…?
Oct
17
revised What is a copresheaf on a “precategory”?
added 484 characters in body
Oct
17
comment Pushout not a homotopy invariant
Indeed, a very similar example works: there's a cospan $1 \rightarrow I \leftarrow 1$ whose pullback is empty, but contracting $I$ to a point first makes the pullback $1$ instead.
Oct
17
comment How do you encode a programm in a category?
You can encode programs as numbers, and you can encode numbers as sets, and you can encode sets as categories, so yes, technically, you can encode programs as categories. This is totally unhelpful, however.
Oct
17
comment Algebraic Varieties, field extensions and tensor product
OK, but what is the definition of $X_K$? Or for that matter, $X \otimes_k K$?
Oct
17
asked What is a copresheaf on a “precategory”?
Oct
17
comment Division by two in set theory
Zero is not a successor ordinal, though.
Oct
17
comment Cardinality of continuous functions on subspace of R^n
Does the codomain need to be Hausdorff for us to conclude that $f = g$ from $f |_D = g |_D$?
Oct
17
comment Algebraic Varieties, field extensions and tensor product
@atricolf: Yes, there is only one $\mathbb{C}$-rational point of $\operatorname{Spec} \mathbb{C}$... but $\operatorname{Spec} \mathbb{C}$ as a scheme over $\operatorname{Spec} \mathbb{R}$ in fact has two $\mathbb{C}$-valued points, corresponding to the fact that $\mathbb{C}$ has a unique non-trivial automorphism over $\mathbb{R}$.
Oct
16
comment Algebraic Varieties, field extensions and tensor product
What definitions are you using? For me, $X_K = X \otimes_k K$ is more-or-less a definition.
Oct
16
comment What's the definition of a free module in an abelian category?
There is no notion of free module that doesn't invoke a forgetful functor to $\textbf{Set}$. The next best thing is, well, a projective object.
Oct
16
revised the limit of infinite product $(1+y)(1+y^2)(1+y^3)(1+y^4)\cdots $
deleted 2 characters in body
Oct
15
comment The opposite category of the category of graphs
If I'm not mistaken, the category of graphs is a locally finitely presentable category but not a preorder, so its opposite cannot be locally finitely presentable. This means there is no hope of finding a description of it as a category of algebraic structures of some kind.
Oct
14
comment Natural transformation monomorphism condition
If a natural transformation is componentwise monic, then it is monic as a natural transformation – this direction is easy. The converse can be proven in several different ways, but you have to use something about the category $\textbf{Set}$. The first one that comes to my mind is to use the fact that $\textbf{Set}$ has equalisers, but this can be done by completely elementary means as well.