Questions: Localization of Categories

Question 1

What do morphisms in the localized category C[W⁻¹] look like, and why can't they simply be morphisms from C with some declared 'inverse'?

• A. Morphisms in C[W⁻¹] are exactly the morphisms in C that were already isomorphisms, plus formal symbols for the inverses of W-morphisms added as new generators
• B. Morphisms in C[W⁻¹] are represented by zig-zags — alternating sequences of forward morphisms from C and backward morphisms formally inverting elements of W — because composing inverses of non-invertible morphisms requires this alternating structure
• C. Morphisms in C[W⁻¹] are the same as in C but with additional equations declaring that W-morphisms compose to identities
• D. Morphisms in C[W⁻¹] are equivalence classes of C-morphisms under the equivalence relation generated by W

Answer: (B)

You cannot simply add formal inverses w⁻¹ for each w ∈ W because composing f·w⁻¹·g for morphisms f, g requires alternating between forward and backward arrows. A zig-zag X → A ← B → C ← D → Y represents a morphism from X to Y in C[W⁻¹], where backward arrows are formal inverses of W-morphisms. Without additional conditions on W (calculus of fractions), such zig-zags may not reduce to manageable forms, and the hom-sets may be proper classes. Option D describes a quotient category, which identifies rather than inverts morphisms — a different construction.

Question 2

The derived category of an abelian category is constructed by localizing which category at which class of morphisms?

• A. The category of abelian groups, localized at all group homomorphisms with trivial kernel
• B. The category of chain complexes, localized at the quasi-isomorphisms — morphisms inducing isomorphisms on all cohomology groups
• C. The category of chain complexes, localized at all chain maps that are split exact
• D. The abelian category itself, localized at all monomorphisms and epimorphisms simultaneously

Answer: (B)

The derived category D(A) of an abelian category A is precisely the localization of the category of chain complexes Ch(A) at the class W of quasi-isomorphisms — morphisms f: A• → B• such that the induced maps H^n(f): H^n(A•) → H^n(B•) are isomorphisms for all n. Two complexes that are quasi-isomorphic look different as complexes but have the same cohomology; in the derived category they become isomorphic. This allows resolutions (injective, projective, flat) to be used interchangeably with the objects they resolve, which is the foundation of derived functors, sheaf cohomology, and homological algebra.

Question 3

Any functor F: C → D that sends every morphism in W to an isomorphism in D factors uniquely through the localization functor γ: C → C[W⁻¹].

• T. True
• F. False

Answer: True

This is the universal property of the localization. The localization C[W⁻¹] is defined to be the category that is initial among all categories where W-morphisms become isomorphisms: any functor F that inverts W must factor through C[W⁻¹] as F = F̄ ∘ γ for a unique functor F̄: C[W⁻¹] → D. This universal property characterizes the localization up to equivalence and is the correct categorical definition — it says C[W⁻¹] is the 'most efficient' way to make W into isomorphisms.

Question 4

Localizing a category C at a class of morphisms W generally produces a category equivalent to some naturally occurring, well-understood category.

• T. True
• F. False

Answer: False

This is explicitly flagged as a misconception in the topic. The result of localization depends sensitively on which W is chosen, and the resulting category may be exotic, ill-behaved, or not equivalent to any previously known category. Worse, without additional conditions on W (such as the calculus of fractions conditions), the hom-sets of C[W⁻¹] may be proper classes, making it not even a legitimate category. The homotopy category (localizing spaces at homotopy equivalences) and derived categories are cases where the result is well-studied, but this is the exception, not the rule.

Question 5

Why is the calculus of fractions condition important for categorical localization, and what goes wrong without it?

Model answer: Without the calculus of fractions conditions on W, morphisms in C[W⁻¹] are represented by arbitrary zig-zags of arbitrary length — alternating forward and backward morphisms — and there is no systematic way to compose or simplify them. The hom-sets become collections of equivalence classes of zig-zags under a complicated equivalence relation, and these collections may be proper classes rather than sets, making C[W⁻¹] ill-defined as a category. The calculus of fractions conditions (closure under composition, common-denominator conditions, and the ore conditions) ensure that every zig-zag can be reduced to a single fraction of the form s⁻¹f or fs⁻¹, giving well-defined, set-sized hom-sets and making composition concrete and manageable.

The calculus of fractions is the condition that allows zig-zags to simplify to fractions — the same reason why, in a ring with a multiplicative set, fractions compose cleanly via common denominators. Without it, the localization may not exist as an ordinary category, and even if it does, working with it is intractable because morphisms do not have canonical representatives.