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Thank you very much.

Also a little bit loud.

Last time I described how quantum field theory on generic curved spacetime

can be defined in terms of a functor. A functor

from the category of spacetimes

to the category of algebras

of observables.

And I showed that this structure implies

structures which had been discussed in physics long ago in terms of

the net of local algebras in the sense of Hagen-Kastler.

Now the formulation in terms of a functor

gives us new concepts by which we can

discuss these theories and the most important concept is the concept of natural transformations.

So one typical question in quantum field theory is when are two theories

equivalent? And often this is expressed in a rather vague way

but here we have a distinguished

concept of equivalence namely this is just a natural equivalence of functors.

So A equivalent to A'

if they are naturally

equivalent in the sense of category theory.

So one would be the equivalence in the sense of physics which has to be

given a definition and the other is the definition within mathematics

and this definition exists and it seems that it covers all what we need for physics.

So what does it say that they are naturally equivalent? It says

there exists

isomorphisms

alpha m from A of m and A' of m

for every space time m such that

the following diagram is commutative so we have

embedding of n into n and then we have a corresponding embedding

of the algebra A of m as a subalgebra of A of n. We have

the isomorphism alpha m

mapping A of m into A' of m. We have

an embedding of A' of m and A' of n.

This was our definition of our

functor and by assumption we have

an isomorphism from A of n into A' of n.

And the naturality of this

equivalence is just the statement that this diagram is commutative.

So whether you make your computations on this side

or on this side this is completely irrelevant

because you can go in both directions. These alpha m's are automorphism

and you can do it locally and then embed your space time in a larger space time

but this will not change anything.

And of course you can also in the same way define

when is a theory a sub theory of another theory. This just would say

that you will replace isomorphism here by homomorphism. It's the same

commutative diagram.

Now there's another