![]() ![]() There’s a way to see the category of finite sets lurking in which we can borrow from this paper: So if you want to generalize, replace by any symmetric monoidal †-category, and replace by the unit for the tensor product. If you want to do this, you’ll need to know that is the unit for the tensor product in We’ll be seeing this guy a lot. So we could, if we wanted, generalize ideas from probability theory this way. One of the rules of the game is that all these equations will make sense in any symmetric monoidal †-category. However, I’m feeling lazy so I’ll often write equations when I could be drawing diagrams. This means we can draw maps using string diagrams in the usual way. We can take tensor products of finitely generated free modules, and this makes into a symmetric monoidal †-category. Do we need to say ‘free’ here? Are there finitely generated modules over that aren’t free?Įvery finitely generated free -module is isomorphic to for some finite set In other words, it’s isomorphic to for some So, is equivalent to the category where objects are natural numbers, a morphism from to is an matrix of numbers in and composition is done by matrix multiplication. Let be the category with of finitely generated free -modules as objects, and module homomorphisms as morphisms. Today we’ll start with the rig of nonnegative real numbers with their usual addition and multiplication let’s call this The idea is that measure theory, and probability theory, are closely related to linear algebra over this rig. It gets more complicated, but also a lot more interesting.īut in fact, a lot still works with a commutative rig, where we can’t necessarily subtract either! Something I keep telling everyone is that linear algebra over rigs is a good idea for studying things like probability theory, thermodynamics, and the principle of least action. But algebraists have long realized that a lot of linear algebra still works with a commutative ring, where you can’t necessarily divide. People often do linear algebra over a field, which is-roughly speaking-a number system where you can add, subtract, multiply and divide. stochastic maps Finitely generated free [0,∞)-modules.What we’ll do is give a unified purely algebraic description of: We could study these questions more generally, and we should, but not today. Let’s restrict attention to probability measures on finite sets, and related structures. (Tobias knows this stuff too, and this is how we think about probability theory, but we weren’t planning to stick it in our paper. They’re sort of buzzing around my brain like flies. I need to write them down now, even if they’re not all vitally important to my paper with Tobias. Last summer-on the 24th of August 2012, according to my notes here-Jamie Vicary, Brendan Fong and I worked through a bunch of these relationships. There are many categories related to probability theory, and they’re related in many ways. So, I might as well write them down here. Also, to get warmed up, I need to think through some things I’ve thought about before. It was a lot of fun I sort of miss that style of working. That earlier paper was developed in conversations on the n-Category Café. It’s a kind of sequel to our paper with Tom Leinster, in which we characterized entropy. I’m trying to finish off a paper that Tobias Fritz and I have been working on, which gives a category-theoretic (and Bayesian!) characterization of relative entropy. ![]()
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