# Additive Structure

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 *p _{X} : X* ®

*Y ^ X*and

*p*®

_{Y}: X*Y ^ Y*, and as the coproduct of

*X*and

*Y*, with injections

*u*®

_{X}: X ^ X*Y*and

*u*®

_{Y}: Y ^ X*Y*. (If

*X = Y*, then our notations for the projections and injections become ambiguous, and we use pi, p

_{2}, ui, u

_{2}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 *A _{X} : X ^ X* ®

*X*, namely the unique morphism whose composites with both projections are the identity morphism

*I*of

_{X}*X*. Dually, as a coproduct, it admits the folding morphism

*V*:

_{X}*X*®

*X ^ X*, whose composites with both of the injections are

*I*. Using the diagonal and folding morphisms, we can define a binary operation, called addition, on Hom(

_{X}*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 0* _{k} S'_{k}* are given by matrices of morphisms between the summands. Specifically, the matrix associated to

*f*:

*®j Sj*^ 0k

*S'*has as its

_{k}*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* : 0*j Sj* ^ 0* _{k} 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

*S*'

_{b}= 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”.