
You Could Have Invented Spectral Sequences (2006) [pdf] - espeed
http://timothychow.net/spectral02.pdf
======
irishsultan
The title made me think of "You Could Have Invented Monads (And Maybe You
Already Have)" ([http://blog.sigfpe.com/2006/08/you-could-have-invented-
monad...](http://blog.sigfpe.com/2006/08/you-could-have-invented-monads-
and.html)), which according to one of the comments in
[http://blog.ezyang.com/2012/02/anatomy-of-you-could-have-
inv...](http://blog.ezyang.com/2012/02/anatomy-of-you-could-have-invented/)
was inspired by this article

~~~
dirkt
Though I have to say that [chain
complexes]([https://en.wikipedia.org/wiki/Chain_complex](https://en.wikipedia.org/wiki/Chain_complex))
and their
[homotopy]([https://en.wikipedia.org/wiki/Homotopy_category_of_chain_com...](https://en.wikipedia.org/wiki/Homotopy_category_of_chain_complexes))
are somewhat more difficult than the structure you need to understand monads.
You can even explain monads purely in programming terms, without reference to
category theory.

~~~
espeed
Here are some illustrations of chain complexes:

Chain Complexes [https://www.mathphysicsbook.com/mathematics/algebraic-
topolo...](https://www.mathphysicsbook.com/mathematics/algebraic-
topology/constructing-surfaces-within-a-space/chain-complexes/)

Homology for Normal Humans
[http://isomorphismes.tumblr.com/post/127950269154/graded-
cha...](http://isomorphismes.tumblr.com/post/127950269154/graded-chain-
complex-of-a-simplex-homology)

------
DennisP
Top of page 2: "Here is a simple example. Suppose we have a chain complex..."

And right there I'm lost because the paper hasn't explained what a chain
complex is, and I don't understand the notation. I guess I couldn't have
invented spectral sequences.

~~~
jordigh
I think this was supposed to be, "if you have an undergraduate education in
mathematics along with the basics of algebraic topology, you could have
invented spectral sequences".

The prototypical example of a chain complex is a triangulation of an
n-dimensional surface. You can take each n-dimensional triangulation into its
(n-1)-dimensional boundary, with the property that the boundary of a boundary
is empty. Roughly, this means that you have a sequence of maps on "chains"
(prototypically, triangulations)

    
    
             d_{n}          d_{n-1}         d_{2}        d_{1}
      C_{n} -----> C_{n-1} --------->  ... -------> C_{1} -------> C_0
    

with the property that d_{k_1}(d_{k})(x) = 0 for all x and all k (or more
abbreviated, d^2 = 0), the boundary of a boundary is zero.

