### Structure and Pseudorandomness

#### Posted by David Corfield

Terence Tao has written three delightful posts, starting here, detailing his views delivered at the Simons’ lectures at MIT on the relationship between structure and pseudorandomness in mathematics. We read

Structured objects are best studied using the tools of algebra and geometry.

Pseudorandom objects are best studied using the tools of analysis and probability.

In order to study hybrid objects, one needs a large variety of tools: one needs tools such as algebra and geometry to understand the structured component, one needs tools such as analysis and probability to understand the pseudorandom component, and one needs tools such as decompositions, algorithms, and evolution equations to separate the structure from the pseudorandomness.

From this position, what do we make of ($n$-)category theory? Is it merely an attempt to deepen our grasp on what is structural in mathematics, and as such it helps us with the whole to the extent that it throws into clearer relief what is pseudorandom?

Just as Tao illustrates hybridness by way of the prime numbers, would it be profitable to view examples of ($n$-)categories as hybrid?

Posted at April 12, 2007 3:40 PM UTC
## Re: Structure and Pseudorandomness

These days I’m spending a lot of time hybridizing $n$-category theory and gauge theory, to get higher gauge theory. To do this hybridization, it’s nice to ‘internalize’ concepts. For example, a (strict) Lie 2-group is a kind of hybrid of a Lie group and a category. Using internalization, we can define it as ‘a category internal to the category of Lie groups’. There are a lot more examples in section 2.1 here. There are also some important subtleties not mentioned here!

Anyway, ‘internalization’ and ‘enrichment’ are two basic methods of forming hybrid concepts in $n$-category theory.