Functors and Limits

Summary

A functor preserves a limit if it sends limit cones to limit cones. It reflects a limit if limit cones in the codomain pull back to limit cones in the domain. It creates a limit if limits in the domain can be constructed uniquely from limits in the codomain. Continuous functors preserve all small limits; right adjoints are always continuous.

Overview

Not all functors interact well with limits. Understanding when a functor preserves, reflects, or creates limits is essential for transferring limit computations between categories. The key theorem — that right adjoints preserve limits — connects functors and limits to the theory of adjoint functors.

Main Content

Definition 5.3.1: Preservation of Limits

A functor $F:\mathcal{A}\to \mathcal{B}$ preserves limits of shape $\mathcal{I}$ if: for every diagram $D:\mathcal{I}\to \mathcal{A}$ and every limit cone $\lambda :\Delta L\Rightarrow D$ in $\mathcal{A}$, the image $F\lambda :\Delta FL\Rightarrow FD$ is a limit cone in $\mathcal{B}$.

Equivalently: $F(\lim D)\cong \lim (FD)$ and the canonical map is an isomorphism.

$F$ is continuous if it preserves all small limits.

Definition 5.3.1: Reflection of Limits

$F$ reflects limits of shape $\mathcal{I}$ if: whenever $F\lambda$ is a limit cone in $\mathcal{B}$, $\lambda$ is a limit cone in $\mathcal{A}$.

Definition 5.3.5: Creation of Limits

$F$ creates limits of shape $\mathcal{I}$ if: for every diagram $D:\mathcal{I}\to \mathcal{A}$ such that $FD$ has a limit in $\mathcal{B}$, there exists a unique cone $\lambda$ over $D$ such that $F\lambda$ is a limit cone, and furthermore $\lambda$ is itself a limit cone.

Hierarchy: Creation implies reflection; preservation is separate (one can preserve without reflecting and vice versa).

Examples

Example: Forgetful Functor Creates Limits (BCT, Ch. 5.3)

The forgetful functor $U:\mathbf{Grp}\to \mathbf{Set}$ creates limits: limits in Grp are computed on the underlying sets and then given the inherited group structure. E.g., the product of groups has underlying set equal to the product of the underlying sets.

This is a general phenomenon: if $\mathcal{A}$ is defined by “sets with structure,” the forgetful functor typically creates limits (because structure is preserved by the set-theoretic limit construction).

Example: Forgetful Functor Does Not Create Colimits (BCT, Ch. 5.3)

The forgetful functor $U:\mathbf{Grp}\to \mathbf{Set}$ does NOT preserve/create coproducts: the coproduct (free product $G\ast H$) has a different underlying set from $UG\bigsqcup UH$.

Lemma on Creating Limits

Lemma 5.3.6 (BCT, Ch. 5.3)

If $F:\mathcal{A}\to \mathcal{B}$ creates limits of shape $\mathcal{I}$ and $\mathcal{B}$ has limits of shape $\mathcal{I}$, then $\mathcal{A}$ has limits of shape $\mathcal{I}$ and $F$ preserves them.

Continuous Functors

Key Fact: Right Adjoints Are Continuous (BCT, Ch. 6.3)

If $F⊣G$, then $G$ preserves all small limits: for any small diagram $D:\mathcal{I}\to \mathcal{B}$ with limit, $G(\lim D)\cong \lim (GD)$.

Dually, left adjoints preserve all small colimits.

This is proved in Adjoints and Limits. It is one of the most useful practical theorems for computing limits.

Examples of the adjoint/continuity theorem:

• $U:\mathbf{Grp}\to \mathbf{Set}$ is a right adjoint (to the free group functor), so it preserves limits.
• $\mathcal{A}(A,-):\mathcal{A}\to \mathbf{Set}$ is a right adjoint (when $\mathcal{A}$ is locally small and has enough structure), so it preserves limits — this is Proposition 6.2.2.

Connections

• The adjoints and limits theorem (Adjoints and Limits) proves that right adjoints preserve limits.
• Continuous functors are exactly the functors that commute with limits, relevant to accessibility and presentability.
• Limits in presheaf categories (Limits in Presheaf Categories): all hom-functors in $[{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}]$ preserve limits.

See Also

• General Limits — What is being preserved/reflected/created
• Colimits — Dual notions
• Adjoints and Limits — Key theorem: right adjoints preserve limits
• Products and Equalizers — concrete limit shapes that functors commonly preserve or create