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Dec
16
comment Need some help with this exercise: why $\operatorname{Aff_n}$ is not closed
It is not true that Aff is the preimage of 1.
Dec
16
comment Terminology question: what does a natural isomorphism do to maps?
If you could have written «It is not an abuse of notation (of language, better) if you préalablement make it clear that you are viewing arrows as objects of such and such category, which is something frequently done.» Abuses of notation or of language occur only in relation to the notational and language conventions established in the context they occur.
Dec
16
comment Terminology question: what does a natural isomorphism do to maps?
You can take the discrete category of all arrows, or the category in which morphisms are just squares with isomorphisms, or that in which maps are restriction and correstriction to subobjects —I've seen the last two used. But that is not my pointc the fact that there are many categories of arrows is irrelevant. That there exists a way to disambiguate an ambiguous statement does not, by itself, remove the ambiguity: you have to make it explicit.
Dec
16
comment Is Map($T^4$,$S^2$) connected?
Well, unless you plan to do it rationally, that is going to be very difficult. Rationally, using minimal models and rational homotopy theory, it's a standard computation.
Dec
15
comment Difficulty understanding “antichains”
As a consequence, if an antichain contains $1$ then $1$ is in fact its only element.
Dec
15
comment Difficulty understanding “antichains”
No. No two elements in an antichain are related.
Dec
15
comment Terminology question: what does a natural isomorphism do to maps?
As you know, Martin, context, context, context.
Dec
15
comment Terminology question: what does a natural isomorphism do to maps?
Well, there is a standard default meaning for the category of groups (which ideally people learn long before even hearing about categories) but you had to link to a Wikipedia page to make clear what caregory of arrows you had in mind. You will not be surprised to learn that quite a lot of people will not be aware of its existance, I guess.
Dec
15
comment Terminology question: what does a natural isomorphism do to maps?
It is if you do not specify the category in which that takees place; there are other categories of arrows where the maps are not isomorphic. That there is a category in which what one says becomes correct does not help with messy writing.
Dec
15
comment Is Map($T^4$,$S^2$) connected?
We re looking at the space of maps $T^2\times T^2\to S^2$, and to such a thing we can attach a bidegree, which will be homotopy invariant. If that works out, we can just contruct two maps with different bidegree (one of bidegree $(1,0)$ and the obvious swap should do the trick) aand that'll show it is not connected.
Dec
15
comment
There is no need to copy your list of badges, which is very easily accessible from your profile page!
Dec
15
comment Mathematical Group for describing the domain of mathematical process
« It is my belief that there is an assumption that any question expressable in english must have a binary answer.» Well, no. «Is it true that the answer of this question is no?» is a simple example. English allows you to ask very weird things, for which it is not sensible to expect an answer making any sense.
Dec
15
comment Homology ring and cohomology ring
The cap product is not an operation inside the homology but an action of the cohomology on homology, strictly speaking. Also, if the space is a smooth closed manifold, then its cohomology is finitely generated as an abelian group, so it cannot be isomorphic to a polynomial ring
Dec
15
comment Homology ring and cohomology ring
If we can assume that then add it to the question.
Dec
15
comment Homology ring and cohomology ring
What is the homology ring? (The ring cannot be isomorphic to a polynomial ring in infinitely many variables and be of rank one in even dimensions at the same time, by the way)
Dec
14
answered Prove that $\frac{(p^{n}-1)(p^{n}-p)…(p^{n}-p^{n-1})}{n!} \in \mathbb{N}$ with $p$ a prime number and $n \in \mathbb{N}$
Dec
13
comment
I think it should be emphasized that it is not the role of moderators to evaluate answers, nor to provide mathematical assistance in elementary or nonelementary subjects.
Dec
13
comment Rings where every subgroup of the additive group is an ideal?
I am pretty sure this has alreaddy been asked.
Dec
13
comment How do I show that $S^1$ is the suspension of $S^0$?
Indeed, the key advice is «draw something». It is hard not to be able to see this if you draw it!
Dec
12
comment Noetherian ring and radical
Then please be explicit, and add somewhere in the body of the request for someone to check what you did.