Fix a set of posts. Its subsets form a category whose morphisms are inclusions that map every element to itself. Call its forgetful functor to Set f. Each BCI can measure its user, such as by producing a vector of neuron activations. Its possible measurements form a space, and these spaces form a category. (Its morphisms would translate between brains, and each morphism would keep track of how well it preserves meaning.) Call its forgetful functor to Set g.
The comma category f/g has as its objects users (each a Set-function from some set of posts they’ve seen to their measured reactions), and each morphism would relate the user to another brain that saw more posts and reacted similarly on what the first user saw.
The product on f/g tells you how to translate between a set of brains. A user could telepathically tell another what headspace they’re in, so long as the other has ever demonstrated a corresponding experience. Note that a republican sending his love for republican posts might lead to a democrat receiving his hatred for republican posts.
The coproduct on f/g tells you how to extrapolate expected reactions between a set of brains. A user could simply put himself into a headspace and get handed a list of posts he hasn’t seen for which it is expected that they would have put him into that headspace.
I’ve found a category-theoretical model of BCI-powered reddit!
Fix a set of posts. Its subsets form a category whose morphisms are inclusions that map every element to itself. Call its forgetful functor to Set f. Each BCI can measure its user, such as by producing a vector of neuron activations. Its possible measurements form a space, and these spaces form a category. (Its morphisms would translate between brains, and each morphism would keep track of how well it preserves meaning.) Call its forgetful functor to Set g.
The comma category f/g has as its objects users (each a Set-function from some set of posts they’ve seen to their measured reactions), and each morphism would relate the user to another brain that saw more posts and reacted similarly on what the first user saw.
The product on f/g tells you how to translate between a set of brains. A user could telepathically tell another what headspace they’re in, so long as the other has ever demonstrated a corresponding experience. Note that a republican sending his love for republican posts might lead to a democrat receiving his hatred for republican posts.
The coproduct on f/g tells you how to extrapolate expected reactions between a set of brains. A user could simply put himself into a headspace and get handed a list of posts he hasn’t seen for which it is expected that they would have put him into that headspace.