Following up on the deep learning and toposes-post, I was planning to do something on the logic of neural networks.
Prepping for this I saw David Spivak’s paper Learner’s Languages doing exactly that, but in the more general setting of ‘learners’ (see also the deep learning post).
And then … I fell under the spell of
Spivak is a story-telling talent. A long time ago I copied his short story (actually his abstract for a talk) “Presheaf, the cobbler” in the Children have always loved colimits-post.
Last week, he did post Poly makes me happy and smart on the blog of the Topos Institute, which is another great read.
If this is way too ‘fluffy’ for you, perhaps you should watch his talk Poly: a category of remarkable abundance.
If you like (applied) category theory and have some days to waste, you can binge-watch all 15 episodes of the Poly-course Polynomial Functors: A General Theory of Interaction.
If you are more the reading-type, the 273 pages of the Poly-book will also kill a good number of your living hours.
Personally, I have no great appetite for category theory, I prefer to digest it in homeopathic doses. And, I’m allergic to co-terminology.
So then, how to define
Any set
This looks like a monomial in a variable
What is
What is
Going from monomials to polynomials we need an addition. We add such representable functors by taking disjoint unions (finite or infinite), that is
If all
The objects in
with all
An object
We can depict

If
Morphisms in
- a map
on the tree-roots in the right direction, and - for any
a map on the branches in the opposite direction
In our manifold/tangentbundle example, a morphism
A smooth map between manifolds
If instead we view the cotangent bundle
But then, I promised not to use co-terminology…
Another time I hope to tell you how