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Sep
26
asked Categories with limits for large diagrams
Sep
26
answered Categories with limits for large diagrams
Sep
26
answered Formalising Statements in Category Theory with Regards to Universes (with an example of the end of a functor)
Sep
26
comment Gaining insight into the Inverse Image Sheaf
It's a filtered colimit – if two open sets contain $f(U)$, then so does their intersection.
Sep
25
revised Hilbert polynomial of product of projective varieties
added 2 characters in body
Sep
25
answered linear algebra proof about kernel
Sep
25
comment Injection and surjection - origin of words
Or, famously, alea iacta est...
Sep
25
comment In what spaces does the Bolzano-Weierstrass theorem hold?
Your question is equivalent to asking, "When is every closed bounded subspace of a metric space compact?" Notice that one necessary condition is the completeness of the metric space – but this is not enough, because the space of sequences with the $\ell^\infty$-norm is complete.
Sep
25
comment Induced Sheaf Structure is equivalent to Inverse Image Sheaf?
You shouldn't think of sections of the structure sheaf as regular "functions" with some specified codomain – in scheme theory, for example, this fails.
Sep
24
comment The existence of ends of functors.
Something like that. The end exists in the enlarged universe $U'$, but you have to show that the end doesn't exist in the original universe $U$ – that is, you must show that the inclusion $\textbf{Set}[U] \hookrightarrow \textbf{Set}[U']$ preserves ends when they exist.
Sep
24
comment How to show a quasi-compact, Hausdorff space be totally disconnected?
Who said anything about $\operatorname{Spec} A$ being discrete? It is a totally disconnected compact Hausdorff space.
Sep
24
comment How to show a quasi-compact, Hausdorff space be totally disconnected?
$[0,1] \cap \mathbb{Q}$ is a totally disconnected compact Hausdorff space that isn't discrete. (Also, Atiyah and Macdonald are two people!)
Sep
24
comment On morphisms on varieties: 1-1 and projective implies iso?
Oh, right – I forgot about that. Thanks for the correction!
Sep
24
comment confusion over the use of universes in category theory
Suppose there are at least two distinct arrows $A \to B$. Take the $\operatorname{mor} \mathcal{C}$-indexed product of $B$ and count the number of morphisms from $A$ to this product.
Sep
23
comment On morphisms on varieties: 1-1 and projective implies iso?
I don't know whether the map you have defined is really a finite map of degree $1$ – I'm just pointing out what you can do if you know that it is!
Sep
23
comment On morphisms on varieties: 1-1 and projective implies iso?
A finite map of degree $1$ between irreducible projective curves is an isomorphism: this is because such a map induces an isomorphism of the field of rational functions on the two curves, and this in turn gives an inverse rational map; but a rational map between two irreducible projective curves is automatically a morphism.
Sep
23
revised confusion over the use of universes in category theory
added 108 characters in body
Sep
23
answered confusion over the use of universes in category theory
Sep
23
comment confusion over the use of universes in category theory
You're reading the wrong definition: a ‘category’ has a set of objects and a set of morphisms; a ‘small category’ has a small set of objects and a small set of morphisms.
Sep
22
comment Unramified functions between Riemann surfaces
First, prove that $F$ is a local homeomorphism. Then show that a surjective local homeomorphism between two compact spaces is automatically a covering map.