Adjoints and Limits

Summary

Right adjoint functors preserve all small limits; left adjoint functors preserve all small colimits. This is one of the most useful theorems in category theory: to show a functor preserves limits, it suffices to find a left adjoint to it. Many concrete preservation results (e.g., hom-functors preserve limits) follow as special cases.

Overview

The connection between adjunctions and limits is the central theorem of Chapter 6: the three subjects of the book (adjunctions, representables, limits) are unified. The adjoint/limit theorem is proved using the Yoneda lemma.

Main Content

Theorem 6.3.1: Right Adjoints Preserve Limits (BCT, Ch. 6.3)

If $F⊣G$ (so $F:\mathcal{A}\to \mathcal{B}$ and $G:\mathcal{B}\to \mathcal{A}$), then:

• $G$ preserves all small limits: $G(\lim D)\cong \lim (GD)$ for any small diagram $D:\mathcal{I}\to \mathcal{B}$.
• $F$ preserves all small colimits: $F(\mathrm{colim}D)\cong \mathrm{colim}(FD)$ for any small diagram $D:\mathcal{I}\to \mathcal{A}$.

Proof (for limits): For any $A\in \mathcal{A}$:

$$\mathcal{A}(A,G(\lim D))\cong \mathcal{B}(FA,\lim D)\cong \underset{I}{\lim }\mathcal{B}(FA,DI)\cong \underset{I}{\lim }\mathcal{A}(A,GDI)\cong \mathcal{A}(A,\lim (GD))$$

By the Yoneda lemma, $G(\lim D)\cong \lim (GD)$.

Examples and Applications

Example: Forgetful Functors Preserve Limits (BCT, Ch. 6.3)

The forgetful functor $U:\mathbf{Grp}\to \mathbf{Set}$ is a right adjoint (to the free group functor $F$), so $U$ preserves all limits. In particular: products of groups have the correct underlying set, the equalizer of group homomorphisms is the set-theoretic equalizer with inherited group structure.

Example: Hom-Functors Preserve Limits (BCT, Prop. 6.2.2)

For any $A\in \mathcal{A}$, $\mathcal{A}(A,-):\mathcal{A}\to \mathbf{Set}$ is a right adjoint (it has a left adjoint when $\mathcal{A}$ has coproducts: $-\bigsqcup A$… actually this is representability). More directly: $\mathcal{A}(A,-)$ is representable/right adjoint in the appropriate sense, hence preserves limits.

Example: Tensor Product Preserves Colimits (BCT, Ch. 6.3)

In $\mathbf{Ab}$ or $R$-$\mathbf{Mod}$, $M{\otimes }_R-$ is a left adjoint (to ${\mathrm{Hom}}_R(M,-)$), so it preserves all colimits. In particular: $M{\otimes }_R(A\oplus B)\cong (M{\otimes }_{R}A)\oplus (M{\otimes }_{R}B)$.

Example: Free Functor Preserves Colimits (BCT, Ch. 6.3)

The free group functor $F:\mathbf{Set}\to \mathbf{Grp}$ is a left adjoint, so it preserves colimits. In particular: $F(X\bigsqcup Y)\cong FX\ast FY$ (free group on a disjoint union is the free product of free groups).

Corollary: Complete Categories

If $\mathcal{A}$ has all small limits, then the limit functor $\lim :[\mathcal{I},\mathcal{A}]\to \mathcal{A}$ is a right adjoint (to $\Delta$), so it preserves limits — limits commute with limits (cf. ^limits-commute).

What Is NOT Preserved

• Right adjoints do not generally preserve colimits. Example: $U:\mathbf{Grp}\to \mathbf{Set}$ does not preserve coproducts (the coproduct of groups is the free product, much larger than the disjoint union of underlying sets).
• Left adjoints do not generally preserve limits.

Connections

• Adjoint functor theorems (Adjoint Functor Theorems): conversely, functors that preserve limits and satisfy a solution-set condition have left adjoints (GAFT), or right adjoints if they preserve colimits (SAFT).
• Continuous functors (Functors and Limits): “right adjoint” implies “continuous.”

See Also

• Adjoint Functors — Adjunction definition
• Functors and Limits — Preservation/reflection/creation
• Adjoint Functor Theorems — Converse direction