Questions: Preservation and Reflection of Limits

Question 1

You discover that functor F: C → D is a right adjoint to some functor G: D → C. Without any further computation, what can you immediately conclude about F?

• A. F preserves all colimits in C
• B. F reflects all limits in D
• C. F preserves all limits that exist in C — including products, equalizers, pullbacks, and terminal objects
• D. F is full and faithful, so it both preserves and reflects all limits

Answer: (C)

The adjoint limit theorem states: right adjoints preserve all limits, left adjoints preserve all colimits. Knowing that F is a right adjoint immediately gives you preservation of every limit type — products, equalizers, pullbacks, terminal objects, and any others. This is one of the most powerful and frequently used results in category theory because it replaces case-by-case verification with a single structural property. Note the distinction: F preserves limits but need not reflect them; and being a right adjoint does not imply F is full or faithful.

Question 2

A functor F: C → D reflects limits. A colleague shows you that F applied to a certain cone λ in C produces a limit cone in D. What can you conclude?

• A. F also preserves limits, since reflecting implies preserving
• B. The cone λ in C was already a limit cone before applying F
• C. Every cone in C maps to a limit cone under F
• D. F is a right adjoint, since reflection of limits is equivalent to being a right adjoint

Answer: (B)

Reflection of limits means: if F(λ) is a limit cone in D, then λ was a limit cone in C. This is the 'backward' direction — it lets you verify limits in C by checking their images in D. This is valuable when D has a more concrete or tractable structure (e.g., D = Set) where limits are easier to verify. Preservation (the forward direction) is independent: a functor can reflect without preserving, or preserve without reflecting. And reflection is not equivalent to being a right adjoint — right adjoints preserve limits, not necessarily reflect them.

Question 3

The hom-functor Hom(A, −): C → Set preserves all limits that exist in C, as a consequence of the interaction between limits and universal properties.

• T. True
• F. False

Answer: True

Hom(A, −) is a right adjoint (to the tensor product or co-product functor, depending on context), and right adjoints preserve all limits. More directly: a limit in C satisfies a universal property that interacts naturally with hom-sets. Formally, Hom(A, lim D) ≅ lim Hom(A, D(−)), meaning maps from A into a limit correspond to compatible families of maps from A into the diagram — exactly preserving the universal property. This fact is used constantly in homological algebra and sheaf theory.

Question 4

A functor that preserves limits is expected to also reflect them — preservation and reflection are equivalent properties.

• T. True
• F. False

Answer: False

Preservation and reflection are logically independent. Preservation says: if λ is a limit cone in C, then F(λ) is a limit cone in D. Reflection says: if F(λ) is a limit cone in D, then λ was a limit cone in C. Neither implies the other. A functor can preserve without reflecting (e.g., a constant functor maps every cone to the same cone, which may happen to be a limit, but this tells you nothing about the original). A functor can reflect without preserving (certain faithful functors detect limits without creating them). The two properties are distinct tools serving different theoretical purposes.

Question 5

State the fundamental adjoint limit theorem and explain why it is practically useful in category theory.

Model answer: Right adjoints preserve all limits; left adjoints preserve all colimits. This is practically useful because it replaces case-by-case verification with a single structural check: if you can show a functor is a right adjoint, you immediately know it preserves products, equalizers, pullbacks, terminal objects, and any other limits — without examining each limit type separately. Conversely, if you want a functor to preserve all colimits, you should make it a left adjoint. Adjoint pairs thus come with automatic limit/colimit guarantees, which is a major reason they are the central structural concept of category theory.

The theorem's utility is seen throughout mathematics. Forgetful functors from algebraic categories (groups, rings, modules) to Set are often right adjoints (to free functors), and this immediately explains why they preserve products and equalizers. Sheafification is a left adjoint, explaining why it preserves colimits. The base change functor in algebraic geometry is a right adjoint, and limit preservation is essential for its role in descent theory. Knowing the adjoint structure of a functor is often the fastest path to its limit-preservation properties.