We begin by requiring A to be an abelian category. This requirement, formulated in detail below, provides a well-behaved addition operation on each of the sets Hom(X, Y), although the requirement is formulated in purely category-theoretic terms and does not explicitly mention this addition operation.

Axiom 1 (Abelian) A is an abelian category. That is

• 1. There is an object 0 that is both initial and terminal. A morphism that factors through this zero object will be called a zero morphism and denoted by 0. Note that each Hom(X, Y) contains a unique zero morphism.
• 2. Every two objects have a product and a coproduct.
• 3. For every morphism a : X ^ Y, thepair a, 0 has an equalizer and a coequalizer. These are called the kernel and cokernel of a.
• 4. Every monomorphism is the kernel of some morphism, and every epimorphism is the cokernel of some morphism.

This axiom has a surprisingly rich collection of consequences, developed in detail in Chap. 2 of [3]. We list here only some of the highlights, which will be important for this paper, and we refer the reader to [3] for the proofs and additional information.

Proposition 1 ([3], Theorem 2.12) Any morphism that is both monic and epic is an isomorphism.

(More generally, as one can easily check, in any category, any equalizer that is an epimorphism is an isomorphism.)

Proposition 2 ([3], Theorem 2.35) The product and coproduct of any two objects coincide.

That is, given two objects X and Y, there is an object X ® Y that serves simultaneously as the product of X and Y, with projections pX : X ® Y ^ X and pY : X ® Y ^ Y, and as the coproduct of X and Y, with injections uX : X ^ X ® Y and uY : Y ^ X ® Y. (If X = Y, then our notations for the projections and injections become ambiguous, and we use pi, p2, ui, u2 instead.) For brevity, we often refer to X ® Y as the sum of X and Y, rather than as the product or coproduct.

As a product, X ® X admits a diagonal morphism AX : X ^ X ® X, namely the unique morphism whose composites with both projections are the identity morphism IX of X. Dually, as a coproduct, it admits the folding morphism VX : X ® X ^ X, whose composites with both of the injections are IX. Using the diagonal and folding morphisms, we can define a binary operation, called addition, on Hom(X, Y) for any objects X and Y. Given f, g : X ^ Y, we define f + g : X ^ Y to be the composite

where f ® g is obtained from the functoriality of products (or of coproducts—they yield the same result).

Proposition 3 ([3], Theorems 2.37 and 2.39) This addition operation makes each Hom(X, Y) an abelian group, with the zero morphism serving as the identity of the group. Composition of morphisms is additive with respect to both factors; that is, when either factor is fixed, the composite f о g is an additive function of the other factor.

Axiom 2 (Vectors) Each of these abelian groups Hom(X, Y) carries an operation of multiplication by complex numbers, making Hom(X, Y) a vector space over C, and making composition of morphisms bilinear over C.

The complex vector spaces Hom(X, Y) will play the role of quantum-mechanical state spaces. For this purpose, they should also be equipped with inner products, making them Hilbert spaces, but, following [9], we refrain from assuming an inner product structure at this stage of the development.[1] It turns out that much of what we shall do later does not depend on the availability of inner products in the vector spaces Hom(X, Y).

An object S in the abelian category A is called simple if S ^ 0 and every monomorphism into S is either a zero morphism or an isomorphism. In other words, S is a non-zero object with no non-trivial subobjects. Because of the abelian structure of A, this definition can be shown (using [3, Theorem 2.11]) to be equivalent to its dual: A non-zero object is simple if and only if it has no non-trivial quotients, i.e., every epimorphism out of S is either a zero morphism or an isomorphism.

Axiom 3 (Semisimple) Every object in A is a finite sum of simple objects.

This axiom considerably simplifies the structure of the vector spaces Hom(X, Y). In the first place, as shown in [3, Sect. 2.3], morphisms from a sum ® ? Sj to another sum 0k S'k are given by matrices of morphisms between the summands. Specifically, the matrix associated to f : ®j Sj ^ 0k S'k has as its a, b entry the composite

Composition of morphisms in A corresponds to the usual multiplication of matrices.

Furthermore, when the summands are simple, we have the following additional information about the matrix entries, a generalization of Schur’s Lemma in group representation theory.

Proposition 4 If f : S ^ S' is a morphism between two simple objects, then f is either the zero morphism or an isomorphism.

Proof The kernel of f is a monomorphism into S, and if it is an isomorphism then f is zero. So, by simplicity of S, we may assume that the kernel of f is zero and therefore (by [3, Theorem 2.17*]) f is a monomorphism. Similarly, by considering the cokernel of f and invoking the simplicity of S', we may assume that f is an epimorphism. But then, by Proposition 1, f is an isomorphism. ?

The last axiom in this subsection combines two flniteness assumptions.

Axiom 4 (Finiteness)

• 1. There are only finitely many non-isomorphic simple objects.
• 2. Each of the vector spaces Hom(X, Y) is finite-dimensional over C.

The first of these two finiteness requirements is merely a technical convenience. The second, however, gives the following important information about the endomor- phisms of simple objects.

Proposition 5 If S is a simple object, then Hom(S, S) = C.

Proof The operation of composition of morphisms is a multiplication operation that makes the vector space Hom(S, S) into an algebra over C. Since S is simple, Proposition 4 says that every non-zero element of this algebra is invertible. That is, Hom(S, S) is a division algebra over C. But C is algebraically closed, so the only finite-dimensional division algebra over it is C itself. ?

Note that the isomorphism Hom(S, S) = C in this proposition can be taken, as the proof shows, to be an isomorphism of algebras, not just of vector spaces. In particular, the identity morphism of S corresponds to the number 1.

Combining this proposition with our earlier observations about matrices, we find that any morphism f : 0j Sj ^ 0k S'k between any two objects in A is given by a matrix whose entries are complex numbers. Moreover, the a, b entry is 0 unless Sb = S'a. From this observation, it easily follows that, when an object X of A is expressed as a sum 0 j Sj of simple objects, the isomorphism types of the summands Sj and their multiplicities are completely determined by X. That is, the representation of X as a sum of simple objects is essentially unique.

• [1] In fact, inner products are never explicitly assumed in [9]. They are, however, implicit in thestatement, in Sect. 5.1 of [9], that certain bases “are - of course - related by aunitary transformation”.