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Jan
14
comment Fibration with CW-complex as basespace admits retraction
It works perfectly. No?
Jan
13
comment Is the existence of finite biproducts a strengthening of commutativity?
It's not so hard to work out if you know some basic facts.
Jan
13
comment Properties of a modified Zariski topology
Well, if you have both $z$ and $\bar{z}$, then you can separate out the real and imaginary parts of $z$. So I would guess that this is the same as the Zariski topology on $\mathbb{R}^{2 n}$.
Jan
13
comment Equivalent definition of Schemes
It is an interesting idea, but it's not at all obvious that schemes in the usual sense have this property.
Jan
13
comment Equivalent definition of Schemes
This is not at all how schemes are defined. Only some colimits define schemes.
Jan
12
comment Frobenius map in scheme theory
There's no difference, because roots of unity are always invertible. The two kernels are isomorphic as $k$-schemes but have different group structures.
Jan
12
comment Show that function $f: A \to B$ is surjective when there is an implication: $g \circ f = h\circ f \to g=h$
Essentially. Go back to the axiom of separation: we have $x \in h (b)$ if and only if $b \in B'$. Of course, $h (b) \subseteq 1 = \{ 0 \}$, so it comes down to $0 \in h (b)$ if and only if $b \in B'$. If $b \notin B'$ then $0 \notin h (b)$, so $h (b) \ne 1$.
Jan
12
comment Open coverings and (co)limits
Yes, see here.
Jan
12
comment Show that function $f: A \to B$ is surjective when there is an implication: $g \circ f = h\circ f \to g=h$
It's an application of the axiom of separation. The only funny thing is that $x$ does not appear on the RHS, but that's allowed.
Jan
12
comment Does the 2-functor $PsAlg\to \mathfrak{X} $ reflect equivalences?
Reflecting isomorphy is not the same as reflecting isomorphisms. Any monadic functor (in the ordinary sense) reflects isomorphisms.
Jan
12
comment Quillen groupoid of a groupoid.
It appears to me that the Quillen groupoid can be described more abstractly as the universal groupoid generated by a category. In which case the answer to your question is yes.
Jan
11
comment Categorical proof that subgroups of free groups are free?
I do not believe there is any such proof. There are any number of similar-sounding statements that are false: for instance, there are rings $R$ such that not every submodule of a free $R$-module is free.
Jan
10
comment Why is $a^nb^n|n\geq1$ not regular and $a^nb^n|n\leq {10^{10}}$ regular?
Of course, in this case, there are $10^{10}$ cases, so the resulting regular expression would be rather long...
Jan
10
comment Coequalizers in categories of relations and partial functions
Indeed $\mathbf{PFun}$ has coequalisers. The easiest way to see this is to show that it is equivalent to the category of pointed sets.
Jan
10
comment Sheafs of abelian groups are the same as $\underline{\mathbb{Z}}$-modules
The constant presheaf and the constant sheaf are related by a universal property. Use it!
Jan
9
comment Structural Induction vs Normal (Mathematical) Induction
No, structural induction cannot always be reduced to mathematical induction. (For example, transfinite induction over the ordinals.) However, mathematical induction is a special case of structural induction.
Jan
8
comment Confused over k-chains and their boundaries.
That is not the correct definition of $\partial C$ in this situation. See the definition of e.g. singular homology.
Jan
7
comment How to calculate row index based on the following ID number?
If there's no pattern the how do you expect to do anything with mathematics? It's not magic!
Jan
7
comment Construction of Yoneda extension
It's a bit difficult to define directly, but it can be done if you really want to. It's easier to describe the right adjoint and describe the left adjoint in terms of that.
Jan
6
comment Why does the fixed point theorem hold for every lambda term?
Very simply: $\lambda$-terms are not the same as numerical functions.