Tom Leinster recently posed an interesting question in a talk at CLAP: “how do I generalize a theorem about objects into a theorem about maps?” The general idea comes from Grothendieck’s relative point of view, and to implement this point of view, one has to overcome certain technical hurdles related to “base change.” I thought I’d spend some time trying to lay out what it means to have a change of basis in algebraic geometry, and then how that idea shows up in Tom’s project: turning entropy into a functor.
You can read about Tom’s project (joint with John Baez and Tobiaz Fritz) directly here: https://ncatlab.org/johnbaez/show/Entropy+as+a+functor
(Currently writing this up, so excuse the notes below!)
I’ll borrow one of Tom’s examples: take a deck of cards, and imagine this as a 13 x 4 matrix, where each entry has 1/52. Let 
*** Information loss example: suppose our domain is a deck of cards represented by a matrix of a 13 x 4 matrix with each entry 1/52. f is a function that reveals the number but not the suit of the card. So it maps that matrix to a vector of 13 matrices, each with entry 1/13. In principle, the information loss of f should be 2 bits: i.e. the information represented by the suits. In the trivial example, if f : (X,p) \to 1, then we should lose H(p) bits!
So let’s keep Tom’s example in the back of our head while we turn to something completely different: scheme theory. Let me start off by quoting the Stacks Project:
One motivation for the introduction of the language of schemes is that it gives a very precise notion of what it means to define a variety over a particular field. For example a variety $latex X$ over $latex Q$ is synonymous with $latex X \to \text{Spec}(Q)$ which is of finite type, separated, irreducible and reduced. In any case, the idea is more generally to work with schemes over a given base scheme, often denoted $latex S$. We use the language: “let $latex X$ be a scheme over $latex S$” to mean simply that X comes equipped with a morphism $latex X \to S$. In diagrams we will try to picture the structure morphism $latex X \to S$ as a downward arrow from $latex X$ to $latex S$. We are often more interested in the properties of $latex X$ relative to $latex S$ rather than the internal geometry of $latex X$. For example, we would like to know things about the fibres of $latex X\to S$, what happens to $latex X$ after base change, and so on.
So a very naive (one might say “ideological”) way of thinking about schemes to is think of them as just objects over other schemes, i.e. objects considered with respect to other objects of the same kind.
Definition 25.18.1. Let S be a scheme.
- We say $latex X$ is a scheme over $latex S$ to mean that $latex X$ comes equipped with a morphism of schemes $latex X \to S$. The morphism $latex X \to S$ is sometimes called the structure morphism.
- If $latex R$ is a ring we say $latex X$ is a scheme over R instead of X is a scheme over Spec(R).
- A morphism f:X→Y of schemes over S is a morphism of schemes such that the composition X→Y→S of f with the structure morphism of Y is equal to the structure morphism of X.
- We denote MorS(X,Y) the set of all morphisms from X to Y over S.
- Let X be a scheme over S. Let S′→S be a morphism of schemes. The base change of X is the scheme XS′=S′×SX over S′.
- Let f:X→Y be a morphism of schemes over S. Let S′→S be a morphism of schemes. The base change of f is the induced morphism f′:XS′→YS′ (namely the morphism idS′×idSf).
- Let R be a ring. Let X be a scheme over R. Let $latex R \to R’$ be a ring map. The base change $latex X_{R’}$ is the scheme
over $latex R’$.
What’s going on here? Without talking about comma categories or anything like that…
Definition: the information loss of a map f:(X,p) \to (Y,s) is L(f) = H(p)-H(s). (Note, it doesn’t depend on the choice of f, just the existence of one!)
Theorem: the following axioms completely characterize the info loss (up to a constant) and therefore the entropy as well.
(1) L(f) = 0 iff f is iso
(2) Series: L(g \circ f) = L(f) + L(g)
(3) Parallel: L(tf \sqcup (1-t)f’) = tL(f) + (1-t)L(f’)
Actually the proof goes through operads. Claim: any meas. pres. f is a convex combination of maps into 1. Given an operad P, we can consider
(1) categorical P-algs
(2) internal algs in a cat P-alg
(3) the free categorical P-alg containing an internal algebra.
For P = 1, (1) is monoidal categories, (2) is monoids, (3) is finite totally-ordered sets.
For P = \Delta_n (the convex space of distributions over a finite space of size n), (1) is R, (2) constant entropy functions, (3) is finite probability spaces.