Bruno Gavranović: belief

Transcript context

…Does 2 morphisms allow us to think about weight tying? Absolutely. So if we think about a parametric morphism as as so it's a map from a to b with a parameter b, we can off we often want to change the parameter space. We often want to say do the weight tying, which, you know, in practice means we start from a smaller weight space and copy the weights in particular. So a 2 morphism in the 2 category of parametric functions is is a reparameterization. So it's a it's a map between 2 parametric morphisms that that is somehow coherent. There are some diagrams that have to be satisfied, but essentially, they encode that 1 is obtained by precomposing some weight some form of weight tying, but it doesn't have to be just copying. Right? That's the thing we're finding out. It can it can be arbitrary relationships between the weights. So 1 of 1 of the things that these 2 cells and 2 morphisms allow us is to see this algebraic structure encoded as relationships between the weights. And then that goes again into what category 3 is about. It's about finding relationships between objects and this. So so absolutely. So here's the key connection to programming. In functional languages, we define data types like lists recursively. A list is either empty or it's an element followed by another list. Categorically, this is an algebra for an endofunctor. The structure map of the algebra packages together all of the constructors of the data type. And the homomorphism from this algebra is exactly what programmers call a fold A function that consumes the list by recursively applying some operation. So the framework is describing the very structure of recursive computation.…

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