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Jun
13
comment Category theoretic approaches to Riemann Hypothesis?
Category theory is mostly a language to express complicated notions. Your question is akin to, "are there any attempts to prove the Riemann hypothesis in German?"
Jun
13
comment Can an algebraic structure such that $x+a=x+b$, have solutions for all $a,b∈K$ exist?
Sure. For instance, take $K = \mathbb{Z}$, interpret $+$ as multiplication, and take $x = 0$.
Jun
13
comment A Function as a collection of Arrows
I would call it the mapping cylinder.
Jun
12
comment Motivation of Strong Monics
Orthogonality is quite useful. For instance, you can prove that (in any category) a morphism that is both strong monic and epic is an isomorphism.
Jun
12
comment A confusion regarding Affine spaces
Affine spaces in algebraic geometry should not be confused with affine spaces in linear algebra.
Jun
12
comment Origins of the name “Q” and “R” for cofibrant and fibrant replacement functors.
Another thought: $R$ stands for reflection, because fibrant replacement can be thought of as the left adjoint to a certain fully faithful functor. (See proposition 4.4.5 in my notes.)
Jun
12
comment Are there categorifications of prime or irreducible elements (of a ring, say)?
This is not so much categorification as it is rephrasing in category-theoretic language.
Jun
11
comment Is there a (hypercomplex) number system, in which addition is **not** commutative
Algebras have, by definition, commutative addition.
Jun
11
comment Zariski vs analytic cohomology of $\mathcal O_X^\times$
Thank you for the kind comments! But all I did was chase references.
Jun
11
comment Is there a recommended symbol for “equal by abuse of notation”?
Put quotation marks around it. A bit typographically difficult, but sure to be understood!
Jun
11
comment Zariski vs analytic cohomology of $\mathcal O_X^\times$
Ah. For me, the Picard group is first defined to be the group of line bundles and then shown to be isomorphic to that $H^1$. Sorry for the confusion!
Jun
11
comment Zariski vs analytic cohomology of $\mathcal O_X^\times$
Sure. But as I said, it suffices to prove that the Picard groups are isomorphic, and line bundles are certainly coherent sheaves.
Jun
11
comment Zariski vs analytic cohomology of $\mathcal O_X^\times$
The proper version (hah!) appears as Théorème 4.4 in [SGA 1, Exposé XII].
Jun
11
comment What is a “connection” in algebraic terms?
I don't think there's any way to get around the use of $\Omega^1_M$ as an external input, because there are too many algebraic Kähler differentials.
Jun
11
comment Zariski vs analytic cohomology of $\mathcal O_X^\times$
The LHS is the algebraic Picard group, the RHS is the analytic Picard group. Doesn't GAGA say these are the same?
Jun
10
comment Weil restriction - from abstract nonsense to a practical procedure
Definition 1 is not quite correct: you should have $L$-morphisms, not $K$-morphisms.
Jun
10
comment Learning the topology needed for topos theory.
@AlistairDermont The amount of background you need is minimal. Even Wikipedia would suffice.
Jun
10
comment Why are frames called “frames”?
They are an abstract axiomatisation of topologies (in the sense of lattices of open sets). A rose by any other name would smell as sweet...
Jun
10
comment How and why did Weierstrass $\wp$ get its special symbol?
It's basically a handwritten Fraktur p. Look up, say, Kurrentschrift or Sütterlin.
Jun
10
comment Learning the topology needed for topos theory.
You can probably just skip the chapter on classical sheaf theory entirely if you have a strong grip on category theory. But any serious mathematician must know at least the definition of topological space, continuous map, etc.