I have a chat room open: Atiyah Macdonald. Whenever I get stuck on something in my notes, I just write the question on there in bold and leave it. Eventually someone comes around and answers it or gives me some good hints, it's been pretty good for me.
it's my favorite cat book so far. another one which was recommended by john baez at uc riverside is : conceptual mathematics: an introduction to categories, which ends up leaving you with a nice bit of topos theory
yeah. I had a buddy of mine email back and forth with him when he was applying to grad schools. the general advice was: people who want to go into cat theory for the sake of it should go to europe, there is not cat theory in us.
:P
there's a funny von neumann anectdote similar to that.
one if his friends saw him talking to his son, he was able to bring himself down to the toddlers level so well, he then thought a bit and realized, von neumann was probably doing the same thing when talking to people in general: bringing himself down do a much lower level.
start with $X$ and define $e_0 :x\mapsto (x,0)$ and $e_1 :x\mapsto (x,1)$. I want the pushout of $X\times [0,1]\leftarrow X \rightarrow X\times [0,1]$ where the maps are $e_i$.
This should be the disjoint union of the cylinders, with bases identified
analytic number theorists like computing. they'd write up huge formulas and huge sums and claim that dis is a mellin transform of a modularform of dat and so on :P
for example, i've heard people say modular forms can be interpreted as sections of vector bundles on modular surfaces. i'm dying to know more about that.
well singular homology is useless for computations, but it's useful because the groups contain all the singular simplices from the get-go, so you don't need to refine and subdivide as you go along
what I'm trying to do today is to learn basic abstract homotopy theory and then incoroprate it conceptually to the homotopy-invariance of singular homology story
okay so we start of with homotopic $f,g$ and we want to show the induced homomorphisms on homology are the same. So we need some kind of analog of homotopy on the level of chain complexes
(just a sec, I'm compiling my notes so I can use them more easily)
Okay, so how does homotopy $f\simeq g$ work in $\mathsf{Top}$? It starts by raising dimension - turning a space $X$ into a "cylinder" $X\times [0,1]$. Second, it provides a way to get back $f$ and $g$ by restricting to the bases of the cylinder
intuitively, each morphism $P_n:S_n(X)\rightarrow S_{n+1}(Y)$ takes a singular $n$-simplex and "thickens it" into a singular prism $P(\sigma):\Delta ^n \times I\rightarrow Y$
we want the restriction to the top base to be the image $g_# (\sigma)$ and the bottom restriction to be $f_# (\sigma)$. We also want a continuous deformation along the height of this prism (cone, really). These things are provided by the homotopy down at the topological level
so we want to take a walk (in the direction "up") on images of the compositions $F\circ (\sigma \times 1_I)$ where $F$ is the homotopy $F:f\simeq g$
Now I need to find a good picture. Do you have a copy of Rotman's algebraic topology to take a look at?
ok, so let's call this prism $P$. It is comprised of the top and bottom bases, and the side faces (it's better to think of those faces as just the "side body" at this level, I think)
okay. Basically, the tensor product is a functor that gives $\mathsf{Ch}_\bullet$ a (closed) monoidal structure. Look up the definition on the nlab real quick
purely informally, it should be an arrow like $X\otimes I\rightarrow Y$ for chain complexes $X,Y,I$, but it's unclear what $I$ should be. It should somehow play the role of an interval
In continuation of this previous question, I'm having problems with the following proposition from Kamps and Porter's Abstract Homotopy and Simple Homotopy Theory:
Proposition (3.7). [page 210]
If $H:X\otimes I \rightarrow Y$ is a homotopy between $f$ and $g$, then defining $h(z)=H(0,...
I haven't been able to work it out in $\mathsf{Ch}_\bullet$, so I'd love some help on that, but conceptually, it's good enough for me for now: internal-hom dictates tensor
ah, but you'd find better advices from me on how to make sausages. anyhow, the bit of number theory i like is algebraic number theory, and modern algebraic NT consists of all sorts of arithmetic geometric stuff, so if you really want to learn number theory, keep sticking to the algebraic side and try understand things geometrically.
a homeomorphism $f : E \to E'$ such that the diagram consisting of $E \to X$, $E' \to X$ and $f : E \to E'$ commutes is called a deck transformation, which i think you're familiar with.
yeah, and now the group of all deck transformations of $E$, a cover of $X$, is called deck transformation group of the covering map $p : E \to X$ which I'll denote by $Aut(E, X)$
in fact, you can prove using a little effort that this correspondence is bijective
this correspondence also has nice properties. if $E' \to E \to X$ are a bunch of regular covers (they're some "nice" covering spaces), then there is a short exact sequence $1 \to Aut(E, X) \to Aut(E', X) \to Aut(E', E) \to 1$
@Exterior i dunno much about it but as far as i know Grothendieck found a way to generalize fundamental groups to algebraic varieties, i.e., totally discrete mental stuff