Representations, omega-representations

Here’s a story:

1. A category $C$ is a gadget that holds information about objects and relationships.
2. A representation of a thing $X$ in a category $C$ is a way to encode $X$ in terms of objects and relationships in $C$.

this is a paradigm that’s instantiated in different settings:

• Thing: a manifold $M$
  • A point is assigned an object
  • A path between two points is assigned a relationship
• Thing: an algebraic scheme $X$
  • A scheme over $X$, $U\to X$, is assigned an object $F(U\to X)$
  • An open map over $X$ is assigned a relationship. (This depends on choice of a topology.)
  • Remark: this relates to ideas we had around higher-topologies for 2-topoi.
  • Rmk: this is a toposic definition that works in any sheaf.
• Thing: a presheaf $X$ over affine schemes
  • Every $Spec(R)\to X$ is assigned an $R$-module.
  • A map of affines over $X$ is assigned a map compatible with base change.
• Thing: stratified/relative/boundary versions of all the above
• Thing: a quiver $Q$
  • A vertex is assigned an object
  • A edge is assigned a relationship.
• Thing: knots/braids
  • Temperly-Lieb category
• Thing: symplectic manifold

Representations of Lie and algebraic groups can be recovered from these, via the existence of classifying spaces & classifying stacks.

We’ll interpret this as, these definitions/constructions specify how a thing $X$ provides instructions for relationships/information flow. Now, very different $X$‘s could end up encoding the same information flow: this is Morita equivalence.

This is useful and practical information: most explicitly, in the case of quivers, a really complicated quiver with 2 million edges could be Morita-equivalent to a quiver with just 5 edges. In fact - it could even be equivalent to the quiver with a single point. Similarly, the knot invariants constructions are designed so that Morita equivalence detects things being the unknot.

Now, Morita equivalence can be relaxed to capture more and more information as we categorify. This adds more and more complexity - but if we add all the complexity at once - categorify infinity-fold at once, there are simplifications/ a different sense of linearity. This is how our new invariants will capture explicit, geometric information about these mathematical objects, and others in nature.