I noticed an informal “recipe” for taking a type of object and constructing invariants (of the object). It’s been useful for removing the feeling of “what, why? where did that come from?” when learning new constructions that fit this recipe. Hopefully it will help you!

- Take in an object
- Look at a collection of structures defined over the object (up to isomorphism)
- Define a binary operation closed over this collection
- Optional: Formally append inverses
- Output: a useful algebraic invariant (used to study the object)

Some examples of this, and comments you are welcome to skip over (you can get a good sense of what I’m saying by just looking at the pictures).

This Burnside ring is the analogue of the representation ring in the category of finite sets, as opposed to the category of finite-dimensional vector spaces (over a field $F$). The Segal theorem *(proved by Gunnar Carlsson, who is a wonderful human being and patient teacher)* is something that I wish to understand, which relates the Burnside ring of a finite group $G$ to the stable cohomotopy of the classifying space $BG$.Let’s fill in the recipe for K-theory:

- Take in a space $M$ (compact Hausdorff to avoid pathologies)
- Look at (complex/real) vector bundles over $M$ (up to iso)
- Equip the collection with the fiberwise direct sum, and the fiberwise tensor product of bundles.
- Formally append rank $-n$ vector bundles
*(a formal entity, defined as an object which, when directly summed with a rank $n$ vector bundle, reduces to a point)* - Output: Topological (complex/real) K-theory

Let’s do an example: What’s the K-theory of a point? Well $\text{Vect}(pt) \simeq \mathbb{N}$, formally appending inverses gives us $K^0(X) \simeq \mathbb{Z}$.

It’s worth stating explicitly the relationship between algebraic and topological K-theory. The algebraic K-theory of (a ring of complex valued $C^\infty$-maps on a space) = the topological $K$-theory of (the space). This, unfortunately, ignores some important subtleties — but it gives you a cartoon to hold on to.

Josh Grochow kindly pointed out that the representation ring and topological K-theory are similar for a good reason! Intuitively, one can think of both as “bundles of representations.”

More specifically, assuming $G$ is finite, one can define the representation ring to be the ($G$-equivariant) K-theory of a point.

$K_G(pt) \simeq R_F(G)$

In other words, finite dimensional linear virtual $F$-representations of $G$ in $R_F(G)$ correspond to virtual equivariant bundles over a point. *Note that the term “virtual” is short for “isomorphism classes of formal differences of” and that $R_F(G)$ is sometimes written as $\text{Rep}(G)$.*

Along similar lines, Jacob Lurie mentions in Loop Spaces, p-Divisible Groups, and Character Theory that

$R_F(G) \to K(BG)$

*(where $K(BG)$ is the topological $K$-theory of the classifying space $BG$ of $G$-principal bundles)* is *almost* an isomorphism, and becomes an isomorphism under p-adic completion.

This fits into the picture of comparing the following two processes:

- completing a $G$-space by making the action free (a geometrical process)
- completing with respect to an ideal (an algebraic process)

Edit: I just began reading Lurie’s Higher Algebra (excerpt below), and it seems that the derived category $\mathcal{D}(R)$ of a ring $R$ fits into ‘the recipe’.

- Take in the ring $R$
- Look at the collection of chain complexes of modules over $R$
- Equip the collection with chain complex homomorphisms (a.k.a chain maps)
- Formally make quasi-isomorphisms into isomorphisms
- Output: The derived category $\mathcal{D}(R)$

Note that the the map $A \to B$ is called a quasi-isomorphism iff the map $H_*(A) \to H_*(B)$ is an isomorphism.

*Thank you to Semon Rezchikov (for explaining to me what a virtual vector bundle was a few weeks ago, it was a lovely and intuitive description that led me to draw the $K$-theory picture above), and to Qiaochu Yuan (for pointing out and correcting the errors in the recipe).*