Quote of the Day: Groupoids

[timelets]

Formulating mathematical reasoning in a language precise enough for a computer to follow meant using a foundational system of mathematics not as a standard of consistency to establish a few fundamental theorems, but as a tool that can be employed in ­everyday mathematical work. There were two main problems with the existing foundational systems, which made them inadequate.

- Firstly, existing foundations of mathematics were based on the languages of predicate logic and languages of this class are too limited.

- Secondly, existing foundations could not be used to directly express statements about such objects as, for example, the ones in my work on 2-theories.

...
The greatest roadblock for me was the idea that categories are “sets in the next dimension.” I clearly recall the feeling of a breakthrough that I experienced when I understood that this idea is wrong. Categories are not “sets in the next dimension.” They are “partially ordered sets in the next dimension” and “sets in the next dimension” are groupoids.

-- V. Voevodsky. https://www.ias.edu/ideas/2014/voevodsky-origins

Watch on YouTube