Comment by HallaSurvivor on Proof that LEM is equivalent to the well-ordering...
Second, topos theory is a pretty formidable subject, and you shouldn't expect to learn it quickly. Getting really familiar with even the basics can take years, rather than months (it certainly took...
View ArticleComment by HallaSurvivor on Is the Axiom of Choice inconsistent with...
@JosephCamacho -- how do you prove that "everyone has a positive (and equal) probability of winning"? What is the probability, say, that person $1$ wins? How would you calculate it?
View ArticleAnswer by HallaSurvivor for Do the Exponent Properties Apply in Constructive...
Yes. In fact, it's quite common to take as an axiom:For any smooth function $f : \mathbb{R}^n \to \mathbb{R}^m$, there is a corresponding function $f : R \to R$ in smooth infinitessimal analysis....
View ArticleAnswer by HallaSurvivor for The importance of subgroups
Subgroups are extremely important to the study of a group -- I really can't overstate that point. I could give dozens of reasons, but here's a few that you might find interesting:If $G$ is complicated...
View ArticleAnswer by HallaSurvivor for Example 2.2 in Rotman's Homological Algebra Text
I think a quick concrete example should (implicitly) answer all three of your questions.Elements of the group ring $kG$ look like polynomials with coefficients in $k$ and variables in $G$. We multiply...
View ArticleAnswer by HallaSurvivor for Why is a $\sigma$-algebra closed only under...
For (1), remember that our goal is to define a measure, which will be a function $\mu : \mathcal{F} \to [0,\infty]$ satisfying$\mu(\emptyset) = 0$$\mu(\bigcup_{n \in \mathbb{N}} A_n) = \sum_{n \in...
View ArticleAnswer by HallaSurvivor for Pushout of $R$-modules.
Write $p : A \oplus B \to A \oplus B \big / \{ (f'm, -g'm) \mid m \in M \}$ for the usual quotient map, sending $(a,b)$ to the coset $(a,b) + \{ (f'm, -g'm) \mid m \in M \}$.Then $\pi_1 : A \to Y$ is...
View ArticleAnswer by HallaSurvivor for Let $R \in \textbf{Ring}$. Show that $A_1...
This is a good exercise, but is confusingly written. Here's a longer version of the same question, along with some hints, which should hopefully clarify the question and help you solve it!For your...
View ArticleAnswer by HallaSurvivor for What is an example of a proof that uses the...
The informal approach to this rule is basically a proof by cases. It's just not obvious from the formal setup.I'm struggling to think of a "natural" example right now, so here's a slightly contrived...
View ArticleAnswer by HallaSurvivor for Why the principle of explosion works?
There are many ways to see why the principle of explosion "should" be true. Here's one which I think should be particularly approachable (even if it's ultimately not "the right way" to see why...
View ArticleAnswer by HallaSurvivor for A subobject is regarded as a proposition -...
Usually we define subobjects of $A$ not as monomorphisms $\phi \hookrightarrow A$, but as equivalence classes of such monomorphisms. Intuitively we should identify two monomorphisms with the same image...
View ArticleAnswer by HallaSurvivor for Prove that the functor $P : \text{Set}^\text{op}...
The first check you should do to see if a functor is representable is whether is preserves limits. Since $P : \mathsf{Set}^\text{op} \to \mathsf{Set}$, that means we want to show $P$ sends limits in...
View ArticleAnswer by HallaSurvivor for "Axioms are the containments in $C$."
For the purposes of this nlab page, an axiom is something of the form "$\Gamma \mid \varphi \vdash \psi$". This is to be interpreted asIn context $\Gamma$ -- if $\varphi$ holds, then $\psi$ holds...
View ArticleAnswer by HallaSurvivor for How Should I Prove this Function is an Isomorphism?
As a hint, you know that $f : M \to N$ is an isomorphism, so it has an inverse $f^{-1} : N \to M$.Can you show that $f^{-1}_* : \text{Hom}(M,L) \to \text{Hom}(N,L)$ is an inverse to $f_*$?I'll leave a...
View ArticleAnswer by HallaSurvivor for Exact meaning of terms in First-Order Logic
A term is something that represents an element of a structure. For instance, if we're working in the language of arithmetic, a term is something like $(1+1)*x + y$. It is built from variables, constant...
View ArticleAnswer by HallaSurvivor for Application of the binomial series: In what point...
It seems like you're happy with the equality:$$\frac{1}{\lVert \vec{r} - \vec{r'} \rVert} = \frac{1}{ \lVert \vec{r} \rVert \sqrt{1+x}}$$where $x = \frac{\vec{r'}^2 - 2 \vec{r} \cdot...
View ArticleAnswer by HallaSurvivor for Why is Möbius function's co-domain $\mathbb{C}$?
Eventually you'll be using these functions to do linear algebra to your posets. For instance, for finite posets we can write the zeta function $\zeta$ as a certain matrix, and then the möbius function...
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Answer by HallaSurvivor for How to do concrete calculations in algebraic...
First, let me give you some broad resources for doing these kinds of computations. Most of them come down to doing manipulations on ideals, and unfortunately that's not a skill that you usually learn...
View ArticleAnswer by HallaSurvivor for How is GNFS the best factoring algorithm when its...
Your algorithm is $O(N^2)$ in the number $N$ we're factoring. But when we talk about complexity of algorithms, we measure with respect to the size of the input, NOT with respect to the size of what the...
View ArticleAnswer by HallaSurvivor for General Topology with Categorical Mindset
I'm surprised nobody has mentioned Bradley, Bryson, and Terilla's Topology: A Categorical Approach. The description sounds like exactly what you're looking for:This graduate-level textbook on topology...
View ArticleAnswer by HallaSurvivor for Recovering a topological space from its...
This is an interesting question, and the quick answer is "no", but the long answer is "kind of". Let me say a bit more.First, there's a quibble here that I don't fully agree with, but I feel obligated...
View ArticleAnswer by HallaSurvivor for Is this construction of a DFA from a NFA wrong?
Great questions!Both (1) and (2) amount to the same confusion, which is how precisely to interpret "the set of all states reachable from some state in $X$ by a $d$-arrow".The idea is that, starting at...
View ArticleAnswer by HallaSurvivor for In a slice category $\textbf{Set}/A$, what can be...
This category is very well understood, since it's once of the canonical examples of a topos. In particular, there are lots and lots of things we can say about it, both abstractly, and via concrete...
View ArticleAnswer by HallaSurvivor for To what extent will Radon-Nikodym's theorem hold...
You can actually prove versions of Radon-Nikodym without AC orLEM! Indeed, there are constructive (in the sense of Bishop) versions of the Radon-Nikodym theorem, though I haven't spent any time...
View ArticleAnswer by HallaSurvivor for Is axiom of replacement nicely stateable in the...
You'll likely be interested in a discussion about replacement that happened in the category theory mailing list a few years ago. You can find it here. Just grep for "replacement" and you'll get to the...
View ArticleAnswer by HallaSurvivor for Is the metric topology determined by its...
I believe this is true -- we can recover what the sequences converge to.Say $(a_n)$ is a sequence in $X$ that we know converges, but we don't know what it converges to. There will be a unique $x \in X$...
View ArticleAnswer by HallaSurvivor for Intuition behind 'adjointness' of adjoint functor...
There are a lot of ways to understand adjoints, because they show up in many places, each of which gives a different perspective on things. For instance, if you're familiar with some galois theory or...
View ArticleAnswer by HallaSurvivor for Is it possible to produce with $3$ elements the...
No.Say you had three elements, $x,y,z$ which generated $G$. Then look at the abelianization$G^\text{ab} = G \big / [G,G]$. The images of $x,y,z$ in this quotient (if you like, $x[G,G], \ y[G,G], $ and...
View ArticleAnswer by HallaSurvivor for How to prove continuity of $g(y)=\max_{x\in K}...
Here's a proof that's more for fun than anything else (especially since there's already a perfectly good answer here).Let $X$ be a sober space (every metric space is sober) and let $K$ be compact and...
View ArticleComment by HallaSurvivor on Giving each student a set of questions so that...
Welcome to mse! Is there a reason you decided to ask this here instead of, say, stackoverflow? It sounds like you're looking for a solution to a coding problem, and your question would likely fit...
View ArticleComment by HallaSurvivor on Interpretation Theorem
I've edited your question to use mathjax (which is searchable) rather than an image (which isn't) so that other users will have an easier time finding this question. In the future, you should do the...
View ArticleComment by HallaSurvivor on Is there a functor from the category of...
A failed attempt, which will hopefully save the next person some time: The pseudofunctor sending a ring $R$ to its category of modules $\mathsf{Mod}_R$ (and sending a ring hom $R \to S$ to the...
View Article"No infinite discrete subspace" vs "No infinite pairwise disjoint family of...
What is the relationship between the two properties (for a topological space $X$)$A$: "$X$ has no infinite family of pairwise disjoint open subsets"$B$: "$X$ has no infinite discrete subspace"We know...
View ArticleComment by HallaSurvivor on "No infinite discrete subspace" vs "No infinite...
Incredible! Thanks for thinking so hard about this, and for summarizing the comments so nicely ^_^
View ArticleComment by HallaSurvivor on Are theorems in mathematics that have only been...
@Z.A.K. -- We're using the fact that topos semantics are complete for (let's say geometric) intuitionistic logic. That is, intuitionistic provability is the same thing as truth in all topoi. If we're...
View ArticleComment by HallaSurvivor on The universal property of subspace topology
@Liam -- No worries at all! Comments are usually quite conversational. You would need to encounter someone particularly rude for them to tell you not to thank someone in the comments! I've left...
View ArticleComment by Chris Grossack on Book recommendations for Combinatorics for...
Knuth, Graham, and Patashnik's Concrete Mathematics for breadth of knowledge, all taught with a computer science leaning. Also Knuth's The Art of Computer Programming, which features lots of...
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