The n-Category Café

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 $x,y,z$, three equations:

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

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

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

and one equation between equations, or syzygy:

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

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

$$a: x \to y$$

$$b: x \to z$$

$$c: y \to z$$

and one 2-dimensional bridge between bridges:

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

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:

$$generators \sim variables \sim islands \sim 0-chains \sim objects$$

$$relations \sim equations \sim bridges \sim 1-chains \sim morphisms$$

$$syzygies \sim equations between equations \sim bridges between bridges \sim 2-chains \sim 2-morphisms$$