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Jun
22
comment Spaces $X$ and $Y$ with $[Z, X]_{\bullet} \cong [Z, Y]_{\bullet}$ for all cogroup objects $Z$ in $\mathsf{hTop}_{\bullet}$
That we have a loop space is irrelevant. This is just an application of the fact that the isomorphisms in $\mathbf{hTop}_*$ are the based homotopy equivalences.
Jun
22
comment Spaces $X$ and $Y$ with $[Z, X]_{\bullet} \cong [Z, Y]_{\bullet}$ for all cogroup objects $Z$ in $\mathsf{hTop}_{\bullet}$
Sure, we could do that too. But the adjointness argument I suggest says that $\Omega f : \Omega X \to \Omega Y$ really is a homotopy equivalence, whether or not $X$ and $Y$ are CW-complexes.
Jun
22
comment Spaces $X$ and $Y$ with $[Z, X]_{\bullet} \cong [Z, Y]_{\bullet}$ for all cogroup objects $Z$ in $\mathsf{hTop}_{\bullet}$
@MichaelAlbanese It seems to me that the answer to Q1 is yes when $X$ and $Y$ are pointed connected CW-complexes. Indeed, by adjointness, we have for every pointed (but not necessarily connected) $Z$ a natural bijection $\mathbf{hTop}_* (Z, \Omega X) \to \mathbf{hTop}_* (Z, \Omega Y)$, so $\Omega f : \Omega X \to \Omega Y$ is a homotopy equivalence, so $f : X \to Y$ is a homotopy equivalence.
Jun
22
comment Spaces $X$ and $Y$ with $[Z, X]_{\bullet} \cong [Z, Y]_{\bullet}$ for all cogroup objects $Z$ in $\mathsf{hTop}_{\bullet}$
@NajibIdrissi Of course, there is also the other option, where the weak equivalences are actual homotopy equivalences.
Jun
22
comment Does mathematics become circular at the bottom? What is at the bottom of mathematics?
There is not a unique choice of foundations in mathematics. Traditionally one starts with some form of predicate calculus, but – here is the important thing – it does not have to be fully formal. You only formalise foundations when you want to study it as a subject in its own right.
Jun
22
comment Does mathematics become circular at the bottom? What is at the bottom of mathematics?
There are some concepts that must be taken as primitive. If you try to understand them in terms of other things then you will indeed end up going in circles.
Jun
22
comment Intuition behind generic point of a scheme?
A generic point in the technical sense is precisely a point whose closure is the whole space. You can think of it as saying that any property that holds on closed subspaces that holds at the generic point must hold everywhere. As for $\operatorname{Spec} \mathbb{R} [x, y] / (x^2 + y^2 - 1)$, you should not be visualising it as a circle at all. (Consider, for instance, $\operatorname{Spec} \mathbb{R} [x, y] / (x^2 + y^2 + 1)$, which has no real-valued points but is nonetheless non-trivial.)
Jun
21
comment Gluing schemes: Tips and tricks.
If you're just checking representability of a functor then you can reduce the cocycle condition (which is a condition on triple intersections) to a somewhat easier condition on double intersections.
Jun
18
comment What's the largest universe we use?
Sometimes I'm lazy and I prove theorems for categories in $U_n$ using categories in $U_{n+1}$.
Jun
18
comment Sub(P) is complete?
That's automatic if you define subfunctors the way I said.
Jun
17
comment Sub(P) is complete?
The relevant definition of subfunctor here is the one where $S (C)$ is literally a subset of $P (C)$ and $s : S (C) \to P (C)$ is the inclusion.
Jun
16
comment short exact sequences of complexes and triangles in the homotopy category
A short exact sequence of complexes goes to a distinguished triangle in the derived category, however.
Jun
13
comment $S$ subring of $R$. Is a projective objects in $R$-$\bmod$ still projective in $S$-$\bmod$?
Well, what about some easy examples, like $\mathbb{Z} \subset \mathbb{Q}$?
Jun
12
comment properties of pullback diagrams
This is called the pullback (pasting) lemma.
Jun
11
comment What is the correct analogue of $\mathbb N$ in a ring of integers?
The obvious map $n \mapsto (n)$ is a bijection between $\mathbb{N}$ and the set of ideals of $\mathbb{Z}$. So perhaps rather than elements one should consider the set of ideals.
Jun
11
comment Points of Weil restriction
The bijection is part of the data of the Weil restriction.
Jun
10
comment Intersection of affine open subschemes
The intersections are always separated: any open subscheme of a separated scheme is separated, after all.
Jun
9
comment Definition of a Cartesian Closed Category
You've written down a bunch of symbols without indicating what they mean. This is hardly a definition.
Jun
9
comment Flatness over tensor product
What if $A = B = M$ is a non-trivial field extension of $k$?
Jun
9
comment Do torsionfree abelian groups form a (possibly many-sorted) algebraic category?
@tcamps The homomorphisms of models of your sketch will have to preserve the $\{ 0 \} \amalg Y$ decomposition, so in particular they will be injective. So you are talking about two different categories of torsionfree abelian groups here.