Homotopical algebra

Homotopical algebra is like playing with building blocks, except these building blocks are really weird and squishy.

You know how sometimes when you build with blocks, you stack them on top of each other and they fit perfectly, but sometimes they don't quite fit right and you have to wiggle them a little bit? Well, with homotopical algebra, we take those wiggles seriously.

We're interested in studying how things fit together, but we allow for some wiggle room – we don't require everything to fit together perfectly on the nose. But we also don't allow for too much wiggle room, because then things wouldn't fit together at all.

So, we have all these squishy building blocks that we can wiggle around and fit together in different ways. And we have some rules about how much we can wiggle them around before they're not allowed to fit together anymore.

We use these rules to study all sorts of things – like shapes and spaces and mathematical objects – and we can learn a lot about how they fit together by wiggling them around with our squishy building blocks.

It might seem weird, but it's actually super useful and helps us understand things that we wouldn't be able to understand otherwise. And it's also a lot of fun to play with squishy building blocks!