Right Adjoints Preserve Limits

Written by: Gabriel Field.

Theorem statement

Perhaps the most frequently used result in category theory is the following.

Theorem. (RAPL)

Let $U : \mathsf{\widetilde{C}} \to \mathsf{C}$ be a right adjoint functor and let $\widetilde{D} : \mathsf{J} \to \mathsf{\widetilde{C}}$ be a diagram with a limit in $\mathsf{\widetilde{C}}$. Then, the composite diagram $U \widetilde{D} : \mathsf{J} \to \mathsf{C}$ has a limit in $\mathsf{C}$ and $$ \lim_{\mathsf{J}} \left( U \widetilde{D} \right) \simeq U \left( \lim_{\mathsf{J}} \widetilde{D} \right) $$ Consequently, if $\mathsf{C}$ and $\mathsf{\widetilde{C}}$ have all $\mathsf{J}$-shaped limits, then the following diagram commutes up to isomorphism.

Diagrammatic proof

Let $F : \mathsf{C} \to \widetilde{\mathsf{C}}$ be a left adjoint and let $\tilde{\lambda} : \tilde{\ell} \Rightarrow \widetilde{D}$ be a limit cone. We can form the cone $U \tilde{\lambda} : U \tilde{\ell} \Rightarrow U \widetilde{D}$, which we wish to show is a limit cone.

To this end, let $\alpha : a \Rightarrow U \widetilde{D}$ be another cone over $U \widetilde{D}$. We wish to show that there is a unique map from the summit $a$ of $\alpha$ to the summit $U \tilde{\ell}$ of our proposed limit cone $U \widetilde{\lambda}$ which commutes with the cones. We first find such a map.

For any $f : i \to j$ in $\mathsf{J}$, the above diagram must commute. Since we have a limit over $\widetilde{D}$, we transpose the left diagram to the right diagram. Commutativity is preserved by adjunctions, so $\overline{\alpha}$ is a cone over $\widetilde{D}$. The cone $\overline{\alpha}$ factors through the limit cone along some (unique) map $\tilde{u}$. Transposing this diagram back, we obtain a map $\overline{\tilde{u}}$ from the cone $\alpha$ to the cone $U \tilde{\lambda}$.

Therefore, there is such a map. Demonstrating its uniqueness is left as an exercise to the reader.

Yoneda-style equational proof

Let $F : \mathsf{C} \to \widetilde{\mathsf{C}}$ be a left adjoint. Then, we have the following isomorphisms, all natural in $z \in \mathsf{C}^{\mathrm{op}}$. $$\begin{align*} \mathsf{C}\left( z, U\left( \lim_{\mathsf{J}} \widetilde{D} \right) \right) &\simeq \widetilde{\mathsf{C}}\left( F z, \lim_{\mathsf{J}} \widetilde{D} \right) \\ &\simeq \lim_{j \in \mathsf{J}} \widetilde{\mathsf{C}}\left( F z, \widetilde{D} j \right) \\ &\simeq \lim_{j \in \mathsf{J}} \mathsf{C}\left( z, U \widetilde{D} \right) \\ \mathsf{C}\left( z, U\left( \lim_{\mathsf{J}} \widetilde{D} \right) \right) &\simeq \mathsf{C}\left( z, \lim_{\mathsf{J}} U \widetilde{D} \right) \end{align*}$$ By the Yoneda lemma, we conclude that $$ \lim_{\mathsf{J}} \left( U \widetilde{D} \right) \simeq U \left( \lim_{\mathsf{J}} \widetilde{D} \right) $$ as desired.[1] [1] This argument is not 100% airtight — it seemingly assumes the existence of $\lim_{\mathsf{J}} U \widetilde{D}$. It is not difficult to fix this inaccuracy, and so it is left as an exercise to the reader.

See also

See also:

Bibliography

  1. Emily Riehl. Category Theory in Context. URL: https://emilyriehl.github.io/files/context.pdf.

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