### The Principle of General Tovariance

#### Posted by Urs Schreiber

Next Friday Klaas Landsman will give a talk here on

The principle of general tovariance:Physical laws should look the same in whatever topos they are defined.

He has made nice slides available online.

In my words, what “tovariance” is supposed to mean is this:

A physical theory should be formulated

internallyin terms of “arrow theory”: a physical theory should be an abstract diagram with certain properties (usually: some parts of it are required to be certain limits or colimits), such that it may be internalized (interpreted) in a suitable ambient category.

“Suitable” ambient categories might in particular be topoi. Or maybe just *Barr exact categories* (as in Glenn’s *Realization of cohomology classes in arbitrary exact categories*) as Igor Baković kindly emphasized to me.

Therefore “the principle of general tovariance” is close to my heart. I have talked about attempts to extract the right arrow theory of quantum field theory of $\sigma$-model type (including all gauge theories) in the series on the charged quantum $n$-particle, which is currently being continued as On BV-Quantization.

In all modesty, I thought of this as an attempt to do for quantum (field) theory what Lawvere has tried to do for classical continuum mechanics: find its right arrow theory. See for instance his Toposes of Laws of Motion which we once discussed a bit here.

Like the principle of general covariance which guided Einstein in the formulation of the classical theory of gravity, the principle of general tovariance becomes rather tautologous after becoming accustomed to it: I think the fact that these ideas once were held as “principles” will in the future just be taken as indication that one had been terribly confused before “discovering” the principle.

Of course the recent ideas by Andreas Döring and Chris Isham on understanding the noncommutativity of the quantum phase space as the result of being internal to a certain topos, which we talked about quite a bit here (I, II, III, IV, V) enters Landsmann’s discussion prominently.

Isham and Döring seem to have seriously sparked new interest in the idea of internalized physics. I wish I could attend the workshop on Categories, Logic and Physics in London next January.

Posted at December 5, 2007 4:18 PM UTC
## Re: The Principle of General Tovariance

There’s a big puzzle surrounding this principle, namely: are we supposed to take our theory of physics as formulated in a

specifictopos, or avariabletopos?The second alternative seems rather far-out, since we normally fix a topos — or more intuitively, a “mathematical universe” — before diving in and formulating a theory of physics. And of course, 99.9% of us fix this topos to be the category of sets!

But, if we decide to work in a

specifictopos, what’s the point of ‘general tovariance’? Sure, it can belogically clearerandhelpful to our thinkingto formulate theories of physics using commutative diagrams, as Urs loves to do. But in general relativity calculations, we really do switch between coordinate systems, so general covariance becomes a very practical thing.In physics, when would we switch between topoi?(Foundations-of-math people will point out that ‘the category of sets’ is not really a fixed topos, since there are different versions of set theory. But, physicists quite rightly doubt whether these various versions of set theory make any difference in concrete problems. The topoi that are

reallyinteresting are the ones that aredrasticallydifferent from any of these ‘categories of sets’.)Perhaps Landsmann’s answer to my big puzzle might be this. We need to formulate our theories of physics to work in a variable topos, because the correct topos is

observer-dependent.We get a little sense of that here:

However, the ‘inner’ topos in this picture is not variable and observable-dependent, but

fixedby our physical theory, which involves a C*-algebra $A$. So, we can still ask: what does formulating our theory in anarbitrarytopos actually buy us here?