The n-Category Café

David wrote:

So we’ve seen, in this thread that, even when we work with nice basic finite sets, the probability monad doesn’t get along too well with logical structure, there being none of that pleasant adjointness between categories of predicates over sets.

Both probability theory and logic are much too good to ‘not get along well’ with each other. They get along fine. It’s merely our job to discover how.

In such fundamental mathematical subjects, nothing can be less than perfect! Any apparent imperfection is merely an imperfection in our own understanding. If we follow the tao, all will be well.

But how do we find the tao? I still think it’s a great project to understand precisely how the probability monad gets along with the ‘logical’ operations on the category of sets (or Polish spaces, or whatever): products, coproducts, more general limits and colimits, exponentials, etcetera.

It would also be nice, if you plan to use Polish spaces for your probability monad, to determine how good the category of Polish spaces actually is. I can’t tell if this category is really ‘right’ or just a sloppy stopgap measure: the first thing somebody happened to try.

For example, is the category of Polish spaces cartesian closed? That is, roughly speaking: does the space of maps between Polish spaces become a Polish space in a nice way? If not, we should switch to something better.

I want to work in a category that’s cartesian closed. In other words — for those who don’t grok this jargon — given spaces $X$ and $Y$, I want to have a space of maps from $X$ to $Y$:

$$Y^X$$

with a natural isomorphism

$$Z^{X \times Y} \cong {(Z^Y)^X}$$

just like we’re used to in the category of sets.

Furthermore, I want a probability measure on this space of maps to give map from $X$ to probability measures on $Y$. In other words, I want a map

$$P(Y^X) \to (P Y)^X$$

Doesn’t that seem sensible?

I don’t know if this map wants to be an isomorphism — I could figure it out in five minutes, but I’ll let someone else. This map should make some diagrams commute… but I don’t know what those diagrams are. Someone who has thought about monads on cartesian closed categories might know.

As mentioned before, I also want a map

$$P X \times P Y \to P(X \times Y)$$

Of course this map should not be an isomorphism, since only product measures are in its range. I bet this map will make $P$ into a symmetric monoidal monad.

Finally (for now), what about

$$P(X + Y)?$$

This one seems sort of funny. There’s not one best map

$$P X + P Y \to P(X + Y)$$

Instead, there’s one natural map for each $p \in [0,1]$: given a probability measure on $X$ and a probability measure on $Y$, you can flip a $p$-weighted coin to pick either $X$ or $Y$, then randomly pick an element from the space you’ve chosen.

All these structures should fit together into some nice entity — which may or may not have been given a name and studied to death.

Hi David. Can I request a “progic iv” post in which you synthesize and linearize projics i-iii and associated comments? It looks very interesting but I find it difficult to follow in its current format; for instance it took me a long time to find the definition of the “probability monad.” Of course it might be a chore.

Cheers,

Progic I-III looks interesting. I sure wish I could read it. MathML, yarghgh!

The work by Carathéodory and Kappos and the relation to von Neumann’s work on quantum probability is explained very nicely by Rota.
[Twelve problems in probability no one likes to bring up

Algebraic Combinatorics and Computer Science: A Tribute to Gian-Carlo Rota
Edition: 1st - Author(s): Rota, Gian-Carlo; Crapo, Henry H.; Senato, D.
2001]

It is his first `problem in probability that no one likes to bring up’.
Scans can be found here:p1p2p3

Recently, Coquand and Palmgren made the connection between the Kappos-Carathéodory point-free measure theory and point-free topology, aka locale theory, topology in a topos. They used metric Boolean algebras, but this can be extended to use Heyting algebras i.e. locales. (I do this in forthcoming work with Coquand).

Lawvere has observed that much of the theory of stochastic processes can be captured using the Kleisli category of the Giry monad. We define this monad on the category of compact regular locales (the formal counterpart of compact Hausdorff spaces). By the Stone-Yosida representation theorem the Riesz space of bounded measurable functions may be represented by continuous functions on a compact regular locale. So, not much seems to be lost by restricting to this category. Combined with Lawvere’s insight, it is not clear to me why it should a priori be difficult to treat stochastic processes formally/algebraically. [Which does not mean that I claim to have done this already.]

To our surprise it turned out that this (commutative!) algebraic treatment of probability theory can be translated to quantum probability by using a different topos. For instance, integrals become states on a C*-algebra and valuations become measures on projections. The details can be found here.

Finally, as I pointed out before the use of valuations on a locale interpreted in a topos extends ordinary probability theory.

To our surprise it turned out that this (commutative!) algebraic treatment of probability theory can be translated to quantum probability by using a different topos. For instance, integrals become states on a C*-algebra and valuations become measures on projections. The details can be found here.

This is the correct link.

Dear Charlie,

The paper by Abramsky is a good start. It is part of a circle of ideas saying that
topology=logic of observations.
Two other related sources are Vickers book:
Topology via Logic
and Smyth’s chapter Topology in Handbook of Logic in Computer Science.

Recently, Steve Vickers wrote a deeper chapter on topology via constructive logic connecting these ideas with topos theory. In short, a Grothendieck topos is a model of a geometric predicate logic.

These aspects (and information systems) play a key role in my paper with Chris Heunen on the application of the internal (and geometric) logic of a topos to Isham’s et.al.’s ideas on topos theory and physics. In that paper we emphasize that geometric logic, as the logic of observations, should also be relevant for physics.

Finally, I expect you know about the book by Segal/Kunze. They continued the development of von Neumann’s ideas on continuous geometries to a complete development of integration theory. References can be found in our paper above.

Bas