The Theory of Topos-Theoretic 'Bridges' – A Conceptual Introduction (2016) | Hacker News Reader

The Theory of Topos-Theoretic 'Bridges' – A Conceptual Introduction (2016)

glass-bead.org

21 pointsebcode4 comments

This is an interesting article, but the absence of simple examples of theories and their toposes and invariants made it seem a little abstract. Surely if the technique is so powerful, there ought to be easy examples of it aplenty.

> Surely if the technique is so powerful, there ought to be easy examples of it aplenty.

Whatever your field of expertise, I suspect that you'll find in it that it is not always true that powerful techniques have any easy examples, let alone plenty of them. (This is definitely true in mathematics, where, often, the prerequisites needed even to define the objects we discuss can be so overwhelming that even many expert mathematicians don't encounter the objects seriously until late graduate school, or afterwards. Every mathematician has their own threshold for when "powerful but difficult" becomes "meaningless abstraction," which, by pure coincidence, usually happens to coincide with the threshold between "used in my research" and "not yet useful to me.")

> Surely if the technique is so powerful, there ought to be easy examples of it aplenty

Some examples of insanely powerful and ubiquitous concepts which do not have easy examples:

- algebraic stacks

- QFT

- _general_ relativity

There's this idea that everything should somehow be explainable to my grandmother, but this idea is never presented with any justification, and it seems to me that there's counterexamples aplenty. And something irks me about the idea. I feel like it comes from the same place as people who've done a 5 minute search on google feeling like they can take on experts who've spent decades studying a subject.

This seems like a needlesly defensive answer. If there are no examples then just say so. It is mathematics after all, some of it is beautiful to explore just for its own sake, no examples or applications neccesary.

But claiming a technique is a bridge between disparate areas of mathematics and then subsequently failing to give concrete examples of such bridging is a bit odd, dont you think?

For what its worth the book "7 sketches in compositionality" (https://arxiv.org/abs/1803.05316) has a chapter on topos theory which provides a good introduction with some simple examples!

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