ABSTRACT. Witness algebra, or

Who needs category theory?

Of course mathematicians do, at least some of them do, because category theory is instrumental in some branches of mathematics, e.g. algebraic topology.

But what about computer scientists or physicists? Do they need category theory?

If category theory is your hammer, some computing problems look like appropriate nails. However the speaker was not impressed and remained skeptical. When he learned that the generally accepted mathematical basis for topological quantum computing is sophisticated category theory, he proposed to his long-time collaborator Andreas Blass to ``decategorize" topological quantum computing.

It turned out, surprisingly, that category theory or something like it is necessary for topological quantum computing. Moreover the root cause of the necessity is not specific to topological quantum computing. There should be numerous other computing problems where something like category theory is necessary. Understanding the root cause allowed us to simplify the mathematical basis for the topological quantum computing and to decategorize it to the extent possible.

In the main part of the talk, without assuming any knowledge of category theory or quantum computing, we illustrate, on a simplified example, why category theory or something like it is necessary for topological quantum computing.