Preservation and Reflection of Limits

Core Idea

A functor F: C → D preserves limits if whenever a diagram in C has a limit cone, F maps it to a limit cone in D. A functor reflects limits if F's image of a cone is a limit in D only when the original cone was a limit in C. Preservation relates to the idea that F respects 'universal' constructions.

Explainer

You know that limits (products, equalizers, pullbacks, terminal objects) are defined by a universal property: a limit cone is the most efficient way to map into a diagram, characterized up to unique isomorphism. You also know that functors are structure-preserving maps between categories. The question of preservation and reflection asks: when you pass a construction through a functor, does the universal property survive?

A functor F: C → D preserves a limit if it maps limit cones to limit cones. Concretely: suppose D: J → C is a small diagram with a limit cone λ: Δ(L) ⇒ D in C (where L is the limit object and each λ_j: L → D(j) is a component of the cone). Then F preserves this limit if the cone F(λ): Δ(F(L)) ⇒ F∘D in D is also a limit cone — meaning F(L) with the maps F(λ_j) satisfies the same universal property in D. Preservation is a statement about what F does: it carries a particular universal construction to another universal construction. The classic example is that the hom-functor Hom(A, −): C → Set preserves all limits that exist in C — this is a consequence of limits and hom-sets interacting via the universal property, and it is one of the most-used facts in category theory.

A functor F reflects a limit if the converse holds: whenever F(λ) is a limit cone in D, the original λ was already a limit cone in C. Reflection is a statement about what you can deduce looking backward through F. If F reflects limits, you can verify that a construction in C is a limit by checking its image in D — a technique used when D has a more concrete or tractable structure. Faithful functors (those injective on hom-sets) often reflect limits, because they don't collapse the morphism information needed to detect universality.

The distinction between preservation and reflection matters for transferring theorems across categories. Right adjoints preserve all limits — this is the fundamental adjoint limit theorem and one of the most useful results in category theory. If F is a right adjoint, you can immediately conclude F preserves products, equalizers, pullbacks, and any other limits. Left adjoints dually preserve colimits. This is why adjoint functors are so powerful: they come with automatic limit-preservation guarantees, letting you compute limits in one category by passing to the other through the adjunction. Knowing whether a functor is an adjoint tells you, in one stroke, what kinds of constructions it respects — and preservation of limits is the primary currency of that respect.