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Apr
8
comment Can the image of chains on a smooth manifold be thought of as a Borel $\sigma$-algebra?
No. Chains are things you integrate over, and for a start, they are not subsets of the manifold.
Apr
8
comment functors on Zero-Object in $_RMod$-category
Well, $F$ would preserve zero objects, but only up to isomorphism.
Apr
8
comment functors on Zero-Object in $_RMod$-category
No. For instance, take a functor that maps every object to $S$ and every morphism to $0$...
Apr
8
comment Exact functors in the category of left R-modules - “Fun for the whole family”
Proceed in several steps. First prove that, in either definition, $T 0 = 0$. Then show that $T$ preserves binary products and kernels (in both definitions). Finally, show that a functor that preserves $0$, binary products, and kernels is left exact (in both definitions).
Apr
8
comment Surjection Vs Surjective geometric morphism
The right definition of "surjective geometric morphism", in my mind, is the one that says $f^*$ is conservative. See Lemma 3 later in the same section.
Apr
8
comment Surjection Vs Surjective geometric morphism
What's wrong with their proof? It's very nice and simple.
Apr
8
comment Can the image of chains on a smooth manifold be thought of as a Borel $\sigma$-algebra?
There is a world of difference between measurable sets and chains! Just because some people choose to write the operations in a $\sigma$-algebra using $+$ and $-$ and so on doesn't mean it's an abelian group under those operations!
Apr
8
comment Do rational functions separate points?
I'm sure varieties are supposed to be separated. :p
Apr
7
comment If $N$ and $M/N$ are free modules of finite rank, so is $M$
No, that's not a valid reason. Why isn't $\mathbb{Z}$ isomorphic to $p \mathbb{Z} \times \mathbb{Z} / p \mathbb{Z}$, then?
Apr
7
comment If $N$ and $M/N$ are free modules of finite rank, so is $M$
Yes, but you need to explain why we have $M \cong N \times M / N$.
Apr
7
comment Cech cohomology and cohomology of a category : a cluster of questions.
I don't know. I'm not sure why one would study the cohomology of categories, actually.
Apr
7
comment Notation for partial function set.
Take a partial function and turn it into a total function by sending elements outside the domain to the new element.
Apr
6
comment Notation for partial function set.
@Student There is a natural bijection between $(T + 1)^S$ and the set of partial functions $S \rightharpoonup T$. Or you can just count.
Apr
6
comment Notation for partial function set.
In some sense, $(T + 1)^S$.
Apr
5
comment The Categories $\mathrm{Set}\times\mathrm{Set}$ and $\mathrm{Set}+\mathrm{Set}$
What makes you say that it has products or coproducts? It does not.
Apr
5
comment Symbolic cancellation in tensor notation of derivative
@beginner The cancellation rule is a little bit more complicated for partial derivatives – the second calculation is incorrect.
Apr
5
comment Is it known if “Homotopy type theory” (HTT) can consistently model objects beyond V?
You seem to be confused about something. Homotopy type theory (abbreviated HoTT, by the way) is not really comparable to ZFC. In the presence of certain axioms, the set-theoretic part of homotopy type theory should have the same strength as, say, Mac Lane set theory plus countably many constants interpreted as distinct inaccessible cardinals.
Apr
5
comment Hausdorff topologies on the natural number set are sigma algebra
You need the complement of every open set to be open, and the only Hausdorff topology with that property is the discrete one.
Apr
4
comment Bijection between closed uncountable sets and R?
You are assuming the continuum hypothesis.
Apr
4
comment The Categories $\mathrm{Set}\times\mathrm{Set}$ and $\mathrm{Set}+\mathrm{Set}$
Your description of $\mathbf{Set} + \mathbf{Set}$ is incorrect. For instance, the objects are pairs $(i, X)$, where $i \in \{ 0, 1 \}$ and $X$ is a set.