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Aug
7
answered Confusion on Cech cohomology
Aug
7
comment Confusion on Cech cohomology
I don't think the bound is supposed to be a good one, just a finite one!
Aug
7
comment Hatcher problem 1.2.3 - technicality in proof of simply connectedness
@BenjaLim Because any convex open subset of $\mathbb{R}^n$ is homeomorphic to $\mathbb{R}^n$. Now remove finitely many points on both sides.
Aug
7
comment Hatcher problem 1.2.3 - technicality in proof of simply connectedness
Your first claim about distinct coordinates is false. Consider the $2^n$ vertices of the unit cube in $\mathbb{R}^n$. Why not try a simpler way of isolating the punctures?
Aug
5
comment Good exercises to do/examples to illustrate Seifert - Van Kampen Theorem
You didn't define what $W_1$ and $W_2$ are. I imagine Hatcher wants you to assume that $W_1$ and $W_2$ are both nice enough for the required deformation retract to exist. For example, if both $X_1$ and $X_2$ are manifolds, then such neighbourhoods exist because a manifold is locally euclidean, and so locally convex.
Aug
5
answered Idempotent complete categories and the Yoneda functor.
Aug
4
revised What is the practical benefit of a function being injective? surjective?
deleted 12 characters in body; edited tags
Aug
4
reviewed Approve suggested edit on Find the probability of the the sum
Aug
4
reviewed Approve suggested edit on What is the difference between set membership symbol $\in$ and $R$?
Aug
4
comment Tropical-like redefinitions of addition and multiplication?
Every distributive lattice is automatically a rig. I dare say the theory of distributive lattices is very rich, as it encompasses, say, boolean algebras, Heyting algebras, pointless topology...
Aug
4
comment Tropical-like redefinitions of addition and multiplication?
@JosephO'Rourke A rig is also known as a semiring. It is what is obtained from the ring axioms when you drop the existence of additive inverses (and add the axiom that $0$ is absorbent for multiplication).
Aug
3
comment Tropical-like redefinitions of addition and multiplication?
The question is vague. To go from the ordinary real numbers to the tropical semiring involves discarding the axiom that additive inverses exist. So what counts as "addition" and "multiplication"? Are you looking for interesting examples of rigs?
Aug
3
revised Find a bijection from $(A^B)^C$ into $A^{B \times C}$
edited tags
Aug
3
comment Find a bijection from $(A^B)^C$ into $A^{B \times C}$
@BrandonK. It denotes two evaluations: $f$ is a function-valued function, so $f(b)$ is itself a function, so it can be evaluated at $c$ to get $f(b)(c)$.
Aug
3
comment Product of quotient map a quotient map when domain is compact Hausdorff?
See this related question and this one.
Aug
2
comment What does the continuum hypothesis imply?
See some of the answers to this MO question.
Aug
2
comment Adjoint to the Hom functor in Boolean rigs
Your latest edit gives a definition that is almost the definition of a distributive lattice. Why not just say that you are thinking about distributive lattices? Nonetheless, you still haven't shown that pointwise operations work – and let me be blunt, they don't. The constant $0$ map is not a homomorphism because it doesn't preserve $1$. So your pointwise "operation" – even if it were well-defined – would not have an identity element.
Aug
2
comment Adjoint to the Hom functor in Boolean rigs
If you are genuinely interested in "structures like a topology, but not a topological space", then you should look at locale theory (a.k.a. pointless topology). Joyal and Tierney give an exposition of the algebraic properties of these objects in their memoir, An extension of the Galois theory of Grothendieck. In particular, they discuss tensor products.
Aug
2
comment Adjoint to the Hom functor in Boolean rigs
You seem to be missing the point. It is true that $\textbf{CMon}$ is a closed category. But that has nothing to do with your "structures like topology", because the category of your "structures" does not embed as a full subcategory. (What's the point of having an extra binary operation if you're not going to preserve it?)
Aug
2
comment Is the intersection of two f.g. projective submodules f.g.?
I tried to construct an algebraic counterexample at first but I got quite confused because the set-theoretic intersection of the vector bundles does not necessarily correspond to the intersection of the corresponding modules of sections. (Try considering the bundles $\operatorname{Spec} k[x,y,z]/(x y - z)$ and $\operatorname{Spec} k[x,y,z]/(x y + z)$ over $\mathbb{A}^1_k$ and see what I mean!)