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Apr
12
comment Inverse limit of small categories
Limits in $\mathbf{Cat}$ are constructed in the obvious way. This is because the functors $\operatorname{ob}, \operatorname{mor} : \mathbf{Cat} \to \mathbf{Set}$ are representable.
Apr
12
comment Natural bijection between $\mathbb{N}$ and algebraic numbers?
The procedure is simple. Fix a well-ordering of polynomials with integer coefficients – to do that, you could take a well-ordering of the integers and use the lexicographic ordering. Then throw away all the ones that are not irreducible. This can all be done computably, maybe even primitive-recursively. Then if you really wanted to, you can numerically approximate the complex roots and then order them lexicographically.
Apr
12
comment Natural bijection between $\mathbb{N}$ and algebraic numbers?
In that case, would you consider the enumeration of polynomials that I suggest to be "natural"?
Apr
12
comment N-Tuples or N-functions in category theory
You can either use products (which is the usual way) or exponential objects (which corresponds to curried functions).
Apr
12
comment Natural bijection between $\mathbb{N}$ and algebraic numbers?
@ThomasAndrews It's not so bad though – you could write a program (for an ideal computer with infinite memory) that enumerates all of them.
Apr
11
comment Base-free proof: Set of generators is Zariski-open
I don't think you can really say you are "writing down equations" if you don't explicitly define $f$...
Apr
11
comment Understanding the isomorphism of Picard group with the first cohomology group
I echo John's comments. Actually, I would say it more strongly: Čech cohomology is the best way of understanding this isomorphism.
Apr
10
comment Base-free proof: Set of generators is Zariski-open
How do you propose to write down equations if you don't have a basis?
Apr
10
comment What is an example of a proof that explicitly relies on the law of excluded middle?
This use of the law of excluded middle can be eliminated: the argument in the "not" branch works just as well.
Apr
10
comment Analogue of locally constant sheaf in algebraic geometry
The short answer to all your questions: go learn about the étale topology, the étale fundamental group, étale cohomology, etc.
Apr
9
comment An example of a space $X$ which doesn't embed in $\mathbb{R}^n$ for any $n$?
Try something very "big".
Apr
9
comment What's up with this endofunctor $\mathbf{Aff}_k \rightarrow \mathbf{Aff}_k$?
I explained that in my answer already. Read carefully and think about what I wrote.
Apr
9
comment Function on a Power Set
Well, you can continue your construction transfinitely...
Apr
9
comment The definition of the $false$ truth value
I have added a paragraph.
Apr
9
comment What's up with this endofunctor $\mathbf{Aff}_k \rightarrow \mathbf{Aff}_k$?
The definition of morphism in (1) is not complete – you should allow the vector space to vary between the domain and the codomain.
Apr
9
comment functors with a morphism lifting property
I don't know any name for that. Grothendieck (pre)fibrations have an additional condition on the lift.
Apr
8
comment functors with a morphism lifting property
Isn't this a discrete fibration?
Apr
8
comment Is there a characterization of coverings in subcanonical pretopologies?
I think some of this is explained in Shulman's paper on exact completions.
Apr
8
comment Is category theory constructive?
No, that's even more inconvenient. Then all the time you have to check that things are cofinally small. It's far from obvious whether, say, the sheaf associated with a cofinally small presheaf is again a cofinally small presheaf. Even the fact that cofinally small presheaves are closed under limits requires a hard theorem. You would know all this if you actually tried to work with these things.
Apr
8
comment How to intrinsically think about simplicial objects.
Yes, things work much better for presheaf toposes.