Additive Categories - Matrix Representation of Morphisms

Matrix Representation of Morphisms

Given objects A₁, … , A_n and B₁, … , B_m in an additive category, we can represent morphisms f: A₁ ⊕ ⋅⋅⋅ ⊕ A_n → B₁ ⊕ ⋅⋅⋅ ⊕ B_m as m-by-n matrices

where

Using that ∑_k i_k ∘ p_k = 1, it follows that addition and composition of matrices obey the usual rules for matrix addition and matrix multiplication.

Thus additive categories can be seen as the most general context in which the algebra of matrices makes sense.

Recall that the morphisms from a single object A to itself form the endomorphism ring End(A). If we denote the n-fold product of A with itself by An, then morphisms from An to Am are m-by-n matrices with entries from the ring End(A).

Conversely, given any ring R, we can form a category Mat(R) by taking objects A_n indexed by the set of natural numbers (including zero) and letting the hom-set of morphisms from A_n to A_m be the set of m-by-n matrices over R, and where composition is given by matrix multiplication. Then Mat(R) is an additive category, and A_n equals the n-fold power (A₁)n.

This construction should be compared with the result that a ring is a preadditive category with just one object, shown here.

If we interpret the object A_n as the left module Rn, then this matrix category becomes a subcategory of the category of left modules over R.

This may be confusing in the special case where m or n is zero, because we usually don't think of matrices with 0 rows or 0 columns. This concept makes sense, however: such matrices have no entries and so are completely determined by their size. While these matrices are rather degenerate, they do need to be included to get an additive category, since an additive category must have a zero object.

Thinking about such matrices can be useful in one way, though: they highlight the fact that given any objects A and B in an additive category, there is exactly one morphism from A to 0 (just as there is exactly one 0-by-1 matrix with entries in End(A)) and exactly one morphism from 0 to B (just as there is exactly one 1-by-0 matrix with entries in End(B)) – this is just what it means to say that 0 is a zero object. Furthermore, the zero morphism from A to B is the composition of these morphisms, as can be calculated by multiplying the degenerate matrices.