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Jan
6
comment Isomorphism of Direct Product of Groups
Also check that it is a homomorphism.
Jan
6
comment Why the kernel of isogeny is finite?
Regarding (2), a discrete compact set is finite. This is easy point-set topology.
Jan
6
comment What do we call a functor which is exact and reflects exact sequences in abelian categories?
I agree with Mariano: I would have said ‘faithfully flat’ as well.
Jan
6
answered Equivalent characterizations of faithfully exact functors of abelian categories
Jan
5
comment What can we say about the map $G\mapsto \text{Aut}(G)$ on the proper class of all groups?
It is a functor if we restrict the domain category to its subcategory of isomorphisms, however.
Jan
5
answered Mnemonic for the fact that a right(left) adjoint functor preserves limits(colimits)
Jan
5
answered Finite generation of ideal in function ring
Jan
5
awarded  Revival
Jan
5
answered Do pushouts preserve regular monomorphisms?
Jan
5
comment Show that the powerset partial order is a cartesian closed category.
That is indeed not-so-well-known. Do you have a reference?
Jan
5
answered Exercise on locally ringed spaces
Jan
3
comment Paradox: Any set theory without universe set is not a model of itself
Any sufficiently powerful set theory cannot possibly contain a model of itself as a set, because then that set theory would prove its own consistency. So, for example, in NF, even though there is a universal set, there is no set that implements the $\in$ relation. (I suspect something even worse is true: for "most" sets $X$ there is no set $R$ such that $R = [\in] \cap (X \times \mathscr{P} X)$.) On the flip side, in ZF there is no universal set, but for every set $X$ there is a set $R$ such that $R = [\in] \cap (X \times \mathscr{P} X)$. So either way you have to give up something.
Jan
3
comment Isomorphism of rings implies isomorphism of vector spaces?
By the way, "finite $k$-algebra" conventionally means a $k$-algebra that is finitely-generated as a $k$-module – in particular it doesn't have to have finitely many elements! (This is the same convention as "finite field extension".)
Jan
3
comment Isomorphism of rings implies isomorphism of vector spaces?
Huh, OK. That's a less interesting question than I thought.
Jan
3
answered Is Klein bottle an algebraic variety?
Jan
3
comment Help understanding Algebraic Geometry
There are too many continuous functions with respect to the Zariski topology. Nonetheless, the Zariski topology is the coarsest topology that makes rational functions continuous. (We put the Zariski topology on the affine line, of course!)
Jan
3
comment Differences between pure and impure set theory?
I'm not sure what you're asking. The construction Clive gives works in any model of ZFU and in particular goes through in ZF. The only difference is that the set of urelements in ZF is always empty, so this only ever produces a trivial automorphism. (When I say "no automorphism", I mean "every automorphism is trivial", which is the same thing for practical non-pedantic purposes.)
Jan
3
comment Flat manifold and metric
The metric for standard spherical coordinates is already diagonal. Do you mean constant?
Jan
3
comment Differences between pure and impure set theory?
@Clive: Actually, any first-order theory with an infinite model also has a model that has a non-trivial automorphism. First, construct an uncountable model, and then observe that there must be an element of the model that is not definable without parameters. Then apply the argument here. The point is that a well-founded model of ZF has no automorphisms, but not every model of ZF is well-founded. (Or perhaps the point is that not every automorphism is definable.)
Jan
3
comment If a functor between categories of modules preserves injectivity and surjectivity, must it be exact?
A normal mono/epimorphism is one that is a kernel/cokernel. An abelian category is precisely an additive category with kernels and cokernels in which all monomorphisms and all epimorphisms are normal.