Welcome to the list Tim. I'm not sure the everything people will be so glad to add Category and Topos Theories among the branch of math which are needed in the search of a TOE :) (They "know" that computability (recursion) theory, provability logics, modal logics, quantum mechanics, etc. can be needed.
But category could be a good move. After all it is a sort of algebraic approach to a mathematical form of Everything ... In my older version of my thesis (a very long french technical report) I have make many sections showing that topos is *the* mathematical structure needed to provide rich models for what I call the first person discourse or the first person universe). This is quasi obvious if you know the relationship between intuitionist logic or Heyting algebra and toposes. Now concerning the third person discourse (the "scientifically communicable" propositions) I have been lead toward symmetric monoidal categories not unlike those used as models of linear logic. Unfortunately I have very strong and non trivial constraints (due to the fact that I use the comp hypothesis in the "philosophy of mind") and I cannot really choose the mathematical structures. Actually I have derived from comp a "theory of physical reality", (Z1*) and I am in search of a semantics for it. It *should* be a braided category in which I could use the Hopf-algebraic tools for solving a (re)normalisation problem in Z1*. In the "new" much shorter version of my thesis I have suppressed all references to categories tough. > Pearl, by the way, the UCLA professor who is the father of the >murdered journalist Danny Pearl (in Pakistan). Gosh! I didn't knew that! I tought PEARL was a common name. >On the subject of "being inside a universe," there are some exciting >papers by Fotini Markopoulou and others on a "category theory" (more >precisely, "topos theory") outlook on this. One of her papers is >"The internal description of a causal set; What the universe looks >like from the inside," 1999. Available at the xxx.lanl.gov arXiv >site as paper gr-qc/9811053. >I'm not ready, yet, to write up my "first posting to the everything >list" about category and topos theory, which are my current main >interests. Well, I guess this obviously _is_ going to be my first >post, by way of Markopoulou's paper dovetailing so directly with Wei >Dai's "being inside a universe" point. I will take a look on Markopoulo's paper. >So I'll at least say what category theory is about. (The book I >recommend is Lawvere and Schanuel's "Conceptual Mathematics: A first >introduction to categories.") A nice elementary introduction indeed. >[SNIP] What's fascinating is that a topos is a kind of "micro >universe.' Not in a physical sense, a la Egan or Tegmark, but in the >sense of generating a consistent reality. More on this later [SNIP]. In the sense of (brouwerian-like) mental construction? Thanks for the references. Regards, Bruno