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April 6, 2007

Cohomology and Computation (Week 19)

Posted by John Baez

We’re continuing our seminar on Classical vs Quantum Computation this spring, but the focus has changed enough that a new title is in order: Cohomology and Computation. I’ll keep up the same numbering system for lectures, though:

  • Week 19 (Apr. 5) - The origin of cohomology in the study of ‘syzygies’, or ‘relations between relations’. Syzygies in the study of linear equations, and more generally in the study of any presentation of any algebraic gadget. Building a topological space from a presentation of an algebraic gadget. Euler characteristic.

Last week’s notes are here; next week’s notes are here.

Note that the picture of a triangle on page 4 is not supposed to precisely match the linear algebra example given earlier. But, they’re similar. In the linear algebra example we had three variables x,y,zx,y,z, three equations:

a:x+y+z0 a: x + y + z \to 0

b:2x+y0b: 2x + y \to 0

c:3x+2y0c: 3x + 2y \to 0

and one equation between equations, or syzygy:

u:a+bc.u: a + b \Rightarrow c .

In the picture of a triangle, we have three 0-dimensional islands x,y,zx,y,z, three 1-dimensional bridges between islands:

a:xya: x \to y

b:xzb: x \to z

c:yzc: y \to z

and one 2-dimensional bridge between bridges:

u:a+bc u: a + b \Rightarrow c

forming a big triangular island.

The island example is simpler than the linear algebra example. I made it simpler so it would be easier to draw. It’s hard to draw — or even imagine — linear combinations of islands! But, in studying the homology of a topological space this is exactly what we do: introduce ‘0-chains’, which are formal linear combinations of islands (that is, points in our space).

You’ll notice the big analogy going on here:

generatorsvariablesislands0chainsobjectsgenerators \sim variables \sim islands \sim 0-chains \sim objects

relationsequationsbridges1chainsmorphismsrelations \sim equations \sim bridges \sim 1-chains \sim morphisms

syzygiesequationsbetweenequationsbridgesbetweenbridges2chains2morphismssyzygies \sim equations between equations \sim bridges between bridges \sim 2-chains \sim 2-morphisms

Posted at April 6, 2007 2:16 AM UTC

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Read the post Cohomology and Computation (Week 20)
Weblog: The n-Category Café
Excerpt: Cohomology and the category of simplices.
Tracked: April 20, 2007 6:20 AM

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