Questions: Abelian Categories

Question 1

Which of the following categories is additive but NOT abelian?

• A. Ab (abelian groups)
• B. R-Mod (modules over a ring)
• C. Free abelian groups
• D. Sheaves of abelian groups on a topological space

Answer: (C)

The category of free abelian groups is additive — hom-sets are abelian groups and biproducts (direct sums) exist — but it fails to be abelian because cokernels of maps between free abelian groups need not be free. For instance, the cokernel of Z →×2 Z is Z/2Z, which is not free. The other three are standard examples of abelian categories.

Question 2

Most Ab-enriched category with biproducts is an abelian category.

• T. True
• F. False

Answer: False

Ab-enrichment plus biproducts makes a category additive, which is only the first layer of abelian. An abelian category additionally requires that every morphism has a kernel and cokernel, and that the canonical map from the coimage to the image is an isomorphism. These exactness conditions are not implied by additivity alone.

Question 3

State what the Freyd-Mitchell embedding theorem guarantees and identify its key restriction.

Model answer: The theorem guarantees that every small abelian category embeds fully, faithfully, and exactly (preserving exact sequences) into R-Mod for some ring R. The key restriction is smallness: the theorem does not apply to large abelian categories such as Ab or R-Mod itself. The practical payoff is that diagram-chasing arguments valid in R-Mod (e.g., the snake lemma proved by element-chasing) transfer to any small abelian category.

The embedding theorem is what licenses the common practice of 'chasing elements' in an abstract abelian category even though its objects may not literally have elements. Without the smallness hypothesis the result fails, so it cannot be invoked for the most common large examples.