### Category Theory and Metaphysics

#### Posted by David Corfield

I have been rather remiss, I feel, in promoting some mutual scrutiny between our $n$-categorical community and that part of the metaphysics community which interests itself in structuralism. Since I heard in the mid-1990s about the metaphysical theory of structural realism put forward by various philosophers of physics, I have thought that category theory should have much to say on the issue. For one thing, a countercharge against those *ontic* structural realists, who believe that all that science discovers in the world are structures, maintains that the very notion of a relation within a structure involves the notion of *relata*, things which are being related. Structures must structure some things. Category theoretic understanding ought to have something to say on this matter.

It’s interesting then to see a recent paper by Jonathan Bain – Category-Theoretic Structure and Radical Ontic Structural Realism – which argues that the countercharge can only be made from a set-theoretic perspective.

So there’s one question: If we adopt the nPOV on physics, can we say what we are committing ourselves to the existence of?

I’ve just had my attention drawn to a related metaphysical question of how to deal with indistinguishables. One proposed solution comes in the shape of quasi-sets. These seem to be an elaborate attempt to graft a notion of indistinguishability onto ZFC with urelements, in order to arrive at collections which are equivalent to other collections when members are replaced by copies of themselves. The intuition here is that a sodium atom is still a sodium atom if we replace one of its electrons by another, and that the particles in a sodium atom form a quasi-set rather than a set.

Setting out from a structural version of set theory, ETCS, where all elements of a set are indistinguishable, we could introduce the notion of different kinds of particle by looking to a similar category of multisets. I see we had some debate at the nLab as to what precisely a multiset should be, and that one side is happy with a version which is equivalent to $Set^2$, where $2$ is the category with two objects and one non-identity arrow. In other words we have a set where members are labelled by members of a second set.

Second question: Is there a place for multisets in the description of a physical system? Would we have to have some functor from Hilbert spaces to Multisets (leaving aside the QFT issue of the relativity of the number of particles)?

## Re: Category Theory and Metaphysics

Two years ago I wrote an essay for the FQXI competition on the nature of time, arXiv:0903.1800, where I described a platonic metaphysics which incorporates the idea of time. Roughly, in the platonic world of ideas there are universes and a universe is considered real or physical if there is a passage of time in it (a good analogy is that an abstract universe is like a DVD of a movie, while the real universe corresponds to the DVD being played in a player). The part where the category theory enters is when one tries to describe the mathematical universes, i.e. a universe that corresponds to some theory of everything (e.g. string theory). My idea was to use the objects and relations among them, which I imagined as labeled graphs. The labels where restricted to be finite series of integers, because a TOE is a theory that has to be described by a finite set of symbols on a finite sheet of paper. However, one can imagine more general universes, where the graph labels are infinite series of integers, which would correspond to theories with an infinite number of axioms, which cannot be described by a finite algorithm.

I have always felt that the whole formalism could be given a category theory reformulation, but I do not have the time nor the expertise to do it.