40,767 reputation
249121
bio website
location
age
visits member for 4 years, 1 month
seen 37 mins ago

Feb
10
comment Reference request - being rigorous about a common abuse of notation.
There is a unique map $\mathbb{N} \to \mathbb{R}$ such that $0$ goes to $0$, $1$ goes to $1$, and $x + y$ goes to $x + y$. That's all you have to say.
Feb
9
comment Is a morphism of $\mathcal O_X$-modules completely determined by the homomorphisms induced on the stalks?
However, it is not true that any collection of morphisms of stalks necessarily comes from a morphism of sheaves...
Feb
9
comment Saying $a \in b$ in category theory
@NickThomas $a \in b$ is not expressible in category theory, for the simple reason that the category cannot distinguish between sets that have the same cardinality.
Feb
8
awarded  Revival
Feb
8
answered Yoga of localization in categories?
Feb
8
comment Reference request: set theory of sigma algebras
For the first statement, use the well-ordering theorem to enumerate $\Omega$, and then just divide it into even and odd parts as you would divide a countable set. (Limit ordinals count as even, of course.)
Feb
8
comment Is there any non-trivial relationship between kernels & kernel pairs?
I would say that the right notion is that of kernel pair, and the fact that kernels work in group theory is nothing more than a coincidence arising from the fact that there is a binary operation $-$ and a constant $0$ such that $x - y = 0$ if and only if $x = y$.
Feb
7
comment $\dim$ $C(X,\mathbb{R})<\infty$ we need to show $|X|<\infty$,
Perhaps it's worth pointing out where we used the compactness assumption...
Feb
7
answered Why is a variety etale locally like affine space?
Feb
7
comment $\dim$ $C(X,\mathbb{R})<\infty$ we need to show $|X|<\infty$,
Can you show that you can always extend a function defined on a finite set of points on $X$ to a continuous function on all of $X$?
Feb
7
comment Is it true that for algebraic sets $V,W$ we have $I(V \times W ) =I(V) + I(W)$?
There is no difference between "variety" (in the sense Martin is using the word) and "algebraic set". This is implicit in the emphasis on reduced rings. If he meant irreducible varieties he would have been talking about integral domains.
Feb
7
comment Manifold definition using sheaves : Is the locally ringed condition necessary?
Do you even need the sheaf of rings? Isn't a topological manifold just a topological space locally homeomorphic to euclidean space?
Feb
6
comment What does projective space classify?
@Andrew If I recall correctly, such a thing defines an embedding into $\mathbb{P}^n$. I'm only asking for morphisms $X \to \mathbb{P}^n$. I've since convinced myself that these should be the same thing as line bundles over $X$ equipped with a chosen embedding into the trivial bundle $X \times k^{n+1}$.
Feb
6
comment Definition of $\text{GL}(n,R)$
We want inverses, so the first definition is definitely the right one.
Feb
6
asked What does projective space classify?
Feb
6
comment Categorial definition of subsets
Regarding the last paragraph, my Galois Theory lecturer tried that and was heavily criticised at the time... but now with hindsight it does make sense!
Feb
5
comment Question about finitely generated projective modules
It is certainly true that every projective module is a direct summand of a free $R$-module.
Feb
5
comment Are faithful functors just monics in Cat?
I don't think it's true that a faithful functor is even monic. For example, any map of sets regarded as a functor is faithful.
Feb
5
comment Is “reflexive transitive closure of relation $R$” a first-order property?
How do we avoid the problem where the model of set theory has a non-standard natural number?
Feb
5
comment Does $f(gh)=gf(h)+f(g)$ make $H^1(G,A)$ small enough?
Actually, if you're just interested in whether it is true that every element of $H^1 (G, A)$ is associated with some $\bar{b}$, then you only need to know that every $G$-module $A$ can be embedded in an injective $G$-module $B$. Then the long exact sequence in cohomology becomes $\cdots \to H^0 (G, B/A) \to H^1 (G, A) \to 0$.