Nate
Reputation
684
Top tag
Next privilege 1,000 Rep.
Create new tags
 Oct 28 comment What is $R(\omega)$ (and where can I find definitions for similar common notation)? Ah, thanks. It helps to know that $R(\omega)$ is called the "rank function". Oct 5 comment Derive adjoint unit from counit Thanks. My issue now is that I have trouble proving that $G_{\epsilon_Y} \circ id_{F(G(Y))}^* = id_{G(Y)}$, not because it's particularly difficult, but because I used this fact to prove the naturality of $id_{F(-)}^*$ so I'm not allowed to invoke naturality in my proof. Instead, I invoked the universal mapping property of the counit to claim that $id_{F(-)}^*$ is $\eta$ (and thus is natural), but I don't think this is quite valid. I understand all the parts of the proof, but I haven't had time to sit down and formalize it yet. Hopefully this answer will be sufficient when I do. Sep 13 comment Equalizers and Basic limit theorem in Category theory Yeah, it seems that $I$, $J$, ... are indeed intended to index the objects $D_I$, $D_J$, ... of $\textbf{D}$. Apologies; this syntax is somewhat new to me. Sep 13 comment Equalizers and Basic limit theorem in Category theory Oh, that was just me being sloppy when summarizing the proof in the book. In the book, $I$ and $J$ are the notation used when referring to elements of $V$ (i.e. only in the indexing of the products), while $D_I$ and $D_J$ are used when referring to objects of $\textbf{D}$ (which are indeed the same, unless I'm missing something). Sorry. I'll go back and edit. Sep 13 comment Equalizers and Basic limit theorem in Category theory @KarlKronenfeld, that's precisely what I was missing, thanks. Care to write up an answer? Sep 13 comment Equalizers and Basic limit theorem in Category theory But limits require that for any limit $X$ with arrow $f_i : X \rightarrow D_i$ in the limit and arrow $g : D_i \rightarrow D_j$ in the diagram, $f_j = g \circ f_i$. If we ignore extra $D_e : D_I \rightarrow D_J$ in the proof then it seems like there are arrows in the diagram (the ignored arrows) that could violate this equation. Aug 3 comment Confusion about Homotopy Type Theory terminology So the section immediately following dependent sums discusses coproduct types, which answers my first question. I'm still curious as to the nature of superexponential types. Are they inhabited by functions of multiple variables? Aug 3 comment Confusion about Homotopy Type Theory terminology Neat. I suppose you could go a step further and say that there's a Successor type data Succ a = Zero | Succ a (known as Maybe to programmers), and then generalize sum types $A + B$ as dependent succession $Succ_{x:A} B$, and then generalize Succession as dependent unity and complete the pattern. Out of curiosity, what do you call independent types that generalize into dependent exponents (i.e. superexponential types), and what would they look like? Aug 3 comment Confusion about Homotopy Type Theory terminology Oh. Neat. I feel silly now. Want to write that up as an answer? Jun 17 comment How do you combine the bipartite function $y=2^{x-1}$ for $x\lt0$ & $y=1-2^{-x-1}$ for $x\ge0$? My confusion was rooted in the fact that the function is not continuously differentiable. Now I feel silly. Jun 12 comment How do you combine the bipartite function $y=2^{x-1}$ for $x\lt0$ & $y=1-2^{-x-1}$ for $x\ge0$? Sure -- but is their a unified equation for the function? It feels like there should be. Jun 10 comment Why do we write second derivatives like $\frac{d^2x}{dt^2}$ This makes very little sense to me. How come we can separate the $d$ from $x$ in $dx$, but we can't separate the $d$ from $t$ in $dt$? What are these $d$ things anyway, and why do we treat the one attached to $x$ differently from the one attached to $t$? May 22 comment Generalizing an alternative derivation of distance @JonasMeyer you're right, it's independent of the cosine rule and just dependent upon the definition of cosine -- I just didn't notice it until I was working through the derivation of the Pythagorean theorem from the cosine rule alone. May 22 comment Generalizing an alternative derivation of distance @MarkDominus No problem. The reason I'm interested is a more-than-600-character rabbit hole. Succinctly, I'm wondering why the Pythagorean theorem occurs. (Why are squares so closely related to measuring distance? What do the squares of the sides have to do with anything?) When I ask, people tend to show me proofs, which is the "how", not the "why", so I'm working through it from another direction. To me it seems that the Pythagorean theorem is emergent from a special case of the cosine rule and that projecting sides onto the axis of distance is a more fundamental way of measuring distance. May 22 comment Generalizing an alternative derivation of distance @QiaochuYuan: d=(√x2+y2) basically says "in order to find d, extrapolate x and y into squares, mush the squares together into one big square, and then measure a side of the new square." I understand why it works, but it seems a coincidental (not fundamental) way to find distance. (Whereas xcos(Y)+ycos(X) is the projection of the x and y axes onto the axis of the distance, that's a very straightforward way of finding it.) Regardless, I'm not asking you to empathize with my curiosity: merely to help me out with my questions. May 22 comment Generalizing an alternative derivation of distance @MarkDominus I'm not asking you to understand why I prefer my approach. May 22 comment Generalizing an alternative derivation of distance @QiaochuYuan it seems unnatural to me that triangular distances are found by extrapolating squares, adding them, and then rooting them. I'm trying to do the same thing but with trig. So by "equivalent" I mean "identical given Cartesian coordinates." By "generalized" I mean this: The Pythagorean theorem works in three-dimensional space. How do you add a z dimension to $x*cos(tan^{-1}(y/x)) + y*cos(tan^{-1}(x/y))$? May 22 comment Generalizing an alternative derivation of distance @MarkDominus I disagree. It depends which one you take as fundamental. Obviously trigonometry and squares are closely related, and obviously you can always transform one into the other, but I'd like to try to derive the Pythagorean theorem from the trig instead of the (more common) inverse. But yes, obviously, the above should reduce to a generalized case of the Pythagorean theorem. May 22 comment Can you show how triangles are related to harmonic growth without using sine? I'm just wondering what happens when you generalize to the case where instead of holding the length of the pole (and the 90* angle with the ground) constant, you hold the length of the opposite angle to the ground constant. It seems that, with this method, all triangles (varying one angle and holding another constant) trace out an ellipse, but I'm not sure how to go about proving or disproving such an idea. May 22 comment Can you show how triangles are related to harmonic growth without using sine? Very slick, I like it. However, the x^2+y^2 = constant part is dependent upon the Pythagorean theorem, and I don't know of any proof that doesn't (at least implicitly) use sine. Is there one? Is there a way to prove that the motion is harmonic even when it's not a right triangle (where, in the example above, you'd explicitly have to use the cosine rule)?