In the category of sets, the product A×B (Cartesian product) and the coproduct A⊔B (disjoint union) are distinct objects with different universal properties. What additional structure in an additive category forces these two constructions to coincide as the direct sum A⊕B?
AThe existence of a terminal object, which forces all limits to equal colimits.
BThe requirement that every morphism is invertible, making products and coproducts trivially isomorphic.
CAb-enrichment: Hom-sets carry abelian group structure and composition is bilinear, enabling the identity i_A∘π_A + i_B∘π_B = id_{A⊕B}.
DThe simultaneous existence of an initial object and a terminal object, whose coincidence defines the direct sum.
The collapse requires Ab-enrichment. When each Hom(A,B) is an abelian group and composition is bilinear, you can write the identity on A⊕B as a sum i_A∘π_A + i_B∘π_B = id. This identity ties together the injection maps (from the coproduct structure) and the projection maps (from the product structure) in a single object. Without the ability to add morphisms, this identity cannot even be stated, and products and coproducts remain separate constructions.
Question 2 Multiple Choice
Which of the following categories is NOT additive?
AThe category Ab of abelian groups
BThe category of vector spaces over a field k
CThe category of R-modules for a commutative ring R
DThe category Grp of all groups (including non-abelian groups)
In Grp, given two homomorphisms f, g: A → B, the pointwise product (f·g)(a) = f(a)·g(a) is generally not a homomorphism when B is non-abelian: (f·g)(ab) = f(a)f(b)g(a)g(b) equals f(a)g(a)f(b)g(b) = (f·g)(a)(f·g)(b) only when f(b) and g(a) commute. So Hom(A,B) lacks a natural abelian group structure in Grp, and the category fails to be Ab-enriched. Ab, Vect_k, and R-Mod are all standard examples of additive categories.
Question 3 True / False
In an additive category, the object A⊕B simultaneously satisfies the universal property of the product A×B (equipped with projection maps) and the universal property of the coproduct A⊔B (equipped with injection maps).
TTrue
FFalse
Answer: True
This is exactly what 'direct sum' means in an additive category — the same object serves both roles. The four maps satisfy π_A∘i_A = id_A, π_B∘i_B = id_B, π_A∘i_B = 0, π_B∘i_A = 0, and i_A∘π_A + i_B∘π_B = id_{A⊕B}. The last identity — which requires adding morphisms — is what ties the product and coproduct structures together into a single object.
Question 4 True / False
Any category with a zero object automatically has the structure needed to form direct sums, since zero morphisms supply the additive identity required for Ab-enrichment of Hom-sets.
TTrue
FFalse
Answer: False
A zero object provides a distinguished zero morphism between any two objects — the additive identity — but a single element is not a full abelian group structure. Ab-enrichment requires every pair of morphisms to have a well-defined sum that is itself a morphism, plus associativity, commutativity, and inverses. The category of pointed sets has a zero object but is not additive. Zero objects are a necessary but far from sufficient condition for additivity.
Question 5 Short Answer
Explain why the category Grp of all groups fails to be additive, even though it has a zero object (the trivial group) and zero morphisms between every pair of objects.
Think about your answer, then reveal below.
Model answer: For Grp to be additive, Hom(A,B) must be an abelian group for every pair A, B — meaning any two homomorphisms f, g: A → B must have a sum f+g that is also a homomorphism. The natural candidate is pointwise multiplication: (f+g)(a) = f(a)·g(a). But for this to be a homomorphism, we need (f+g)(ab) = (f+g)(a)·(f+g)(b), which expands to f(a)f(b)g(a)g(b) = f(a)g(a)f(b)g(b). This holds only when f(b) and g(a) commute for all a, b — precisely when B is abelian. For a non-abelian target B, the pointwise product of two homomorphisms is generally not a homomorphism, so Hom(A,B) has no natural abelian group structure and Grp is not Ab-enriched.
The key insight is that Ab-enrichment is a condition on Hom-sets, not just on objects. The full subcategory Ab ⊂ Grp of abelian groups IS additive for exactly this reason — restricting to abelian targets makes pointwise addition of homomorphisms well-defined.