Grant -

[Categorization](https://en.wikipedia.org/wiki/Categorization) and [categorification](https://en.wikipedia.org/wiki/Categorification) are completely different things, at least if people are using words correctly. People get confused sometimes. But it's quite possible Ghrist was using "categorization" to mean what it really means: the job of taking things and classifying them into different kinds, e.g. writing a program that can tell the difference between pictures of cats and pictures of dogs. Applied topology is sometimes used for things like this.

An operad is not like a simplicial complex. A simplicial complex is a space made out of simplices:



An operad is a gadget with a set of "n-ary operations" \\(O_n\\) for each \\(n = 0,1,2,\dots\\). For example, if you take \\(O_n\\) to have \\(n!\\) elements, it can describe all the ways you can take \\(n\\) elements in a monoid or ring and multiply them (in all possible orders).

I explained the more general 'typed operads' here:

* [Complex adaptive system design (part 3)](https://johncarlosbaez.wordpress.com/2017/08/17/complex-adaptive-system-design-part-3/)

and you get back to the ordinary operads if you take the set of types to have 1 element. We use typed operads in our CASCADE project to describe ways of assembling networks of various kinds of agents. If you have two kinds of agents - say, cars and trucks - then your set of types has two elements.