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Feb
4
comment Is the category of sheaves on a site always abelian?
It's certainly true in the special case where $\mathcal{A} = \textbf{Ab}$, but why do you want it in this generality?
Feb
4
answered Existence of a functor $\mathsf{Sets} \to \mathsf{Groups}$ that admits a left adjoint
Feb
3
comment What group does $\mathbb{G}_m$ denote?
In fact, $F$ need not even be a field but just any commutative ring...
Feb
3
comment Isomorphism between spaces of sections.
Well, obviously, $\Gamma^0 (E_1)$ is isomorphic to $\Gamma^0 (E_2)$ if $E_1$ and $E_2$ are isomorphic. Less obviously, if you know $\Gamma^0 (E_1)$ and you know the action of $C^0(M, \mathbb{R})$ on $\Gamma^0 (E_1)$, then you can reconstruct $E_1$ up to isomorphism.
Feb
3
comment Two irreducible plane affine curves over an algebraically closed field $k$ are homeomorphic under the Zariski topologies
Because $k$ is an infinite field, and for all infinite cardinals $\kappa$, $\kappa = \kappa^2$. (This requires the axiom of choice.)
Feb
3
awarded  Custodian
Feb
3
reviewed Satisfactory An entire function with two periods
Feb
2
awarded  Popular Question
Feb
2
answered Terminology: Sheaves with surjective structure maps?
Feb
2
answered What are the $2$-morphisms in the $2$-category of “categories over $\mathfrak{Sch}$”?
Feb
2
revised definition of left (right) Exact Functors
added 1 characters in body
Feb
1
comment The ultraproduct $\mathbb{N}^\mathbb{N} / \mathcal{F}$ is uncountable
@BrianM.Scott Actually, one could also weasel out by saying that if there are no non-trivial ultrafilters, then the proposition is vacuously true...
Jan
31
comment The relation between retraction and coproduct in R-Mod
Oops, I think I must have misread the question... as stated this is true in any abelian category, as t.b. explained here.
Jan
30
comment The relation between retraction and coproduct in R-Mod
It's true in the category of vector spaces over a field, but I suspect not in general.
Jan
30
comment Does there exist another way of obtaining a topological space from a metric space equally deserving of the term “canonical”?
You really must specify what morphisms you are using in $\textbf{Met}$, otherwise the question is not well-posed.
Jan
30
answered Equivalent Definitions of Types
Jan
30
comment Proof that $B^{A \times A'}$ is isomorphic to $(B^A)^{A'}$ in a CCC
$\textrm{Hom}(X, -)$ doesn't reflect isomorphisms in general, but the collection of all $\textrm{Hom}(X, -)$ jointly reflect isomorphisms.
Jan
29
comment classical topology but with lattices
If you demand that your frame $\tau$ is embedded as a subframe of $P X$ for some set $X$, then you are asserting that $\tau$ actually is a topology on $X$. In effect, you are suggesting that we restrict the study of locales to those that actually come from topological spaces. In other words there is no added generality at all! So why not just study these things qua topological spaces?
Jan
28
comment Finding a functor satisfying a recursive equation
Perhaps Kelly's 1980 paper, A unified treatment of transfinite constructions, is relevant here.
Jan
27
answered Images of simple modules under exact endofunctors