In general relativity, we represent possible universes using relativistic spacetimes, which are Lorentzian manifolds (M, g), where M is a smooth four dimensional manifold, and g is a smooth Lorentzian metric. An isometry between spacetimes (M,g) and (M,g′) is a smooth map φ : M → M′ such that φ ∗ (g′) = g, where φ∗ is the pullback along φ. We do not require isometries to be diffeomorphisms, so these are not necessarily isomorphisms, i.e., they may not be invertible. Two spacetimes (M,g), (M′,g′) are isomorphic, if there is an invertible isometry between them, i.e., if there exists a diffeomorphism φ : M → M′ that is also an isometry. We then say the spacetimes are isometric.
The use of category theoretic tools to examine relationships between theories is motivated by a simple observation: The class of models of a physical theory often has the structure of a category. In what follows, we will represent general relativity with the category GR, whose objects are relativistic spacetimes (M,g) and whose arrows are isometries between spacetimes.
According to the criterion for theoretical equivalence that we will consider, two theories are equivalent if their categories of models are “isomorphic” in an appropriate sense. In order to describe this sense, we need some basic notions from category theory. Two (covariant) functors F : C → D and G : C → D are naturally isomorphic if there is a family ηc : Fc → Gc of isomorphisms of D indexed by the objects c of C that satisfies ηc′ ◦ Ff = Gf ◦ ηc for every arrow f : c → c′ in C. The family of maps η is called a natural isomorphism and denoted η : F ⇒ G. The existence of a natural isomorphism between two functors captures a sense in which the functors are themselves “isomorphic” to one another as maps between categories. Categories C and D are dual if there are contravariant functors F : C → D and G : D → C such that GF is naturally isomorphic to the identity functor 1C and FG is naturally isomorphic to the identity functor 1D. Roughly speaking, F and G give a duality, or contravariant equivalence, between two categories if they are contravariant isomorphisms in the category of categories up to isomorphism in the category of functors. One can think of dual categories as “mirror images” of one another, in the sense that the two categories differ only in that the directions of their arrows are systematically reversed.
For the purposes of capturing the relationship between general relativity and the theory of Einstein algebras, we will appeal to the following standard of equivalence.
Theories T1 and T2 are equivalent if the category of models of T1 is dual to the category of models of T2.
Equivalence differs from duality only in that the two functors realizing an equivalence are covariant, rather than contravariant. When T1 and T2 are equivalent in either sense, there is a way to “translate” (or perhaps better, “transform”) models of T1 into models of T2, and vice versa. These transformations take objects of one category – models of one theory—to objects of the other in a way that preserves all of the structure of the arrows between objects, including, for instance, the group structure of the automorphisms of each object, the inclusion relations of “sub-objects”, and so on. These transformations are guaranteed to be inverses to one another “up to isomorphism,” in the sense that if one begins with an object of one category, maps using a functor realizing (half) an equivalence or duality to the corresponding object of the other category, and then maps back with the appropriate corresponding functor, the object one ends up with is isomorphic to the object with which one began. In the case of the theory of Einstein algebras and general relativity, there is also a precise sense in which they preserve the empirical structure of the theories.