nLab adjoints preserve (co-)limits

Idea

One of the basic facts of category theory is that left/right adjoint functors preserves co/limits, respectively.

Statement

Proposition

Let $\mathcal{C}$ and $\mathcal{D}$ be two categories and let

$$(L \dashv R) \;\colon\; \mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot} \mathcal{D}$$

be a pair of adjoint functors between them.

Then

1. If $X \colon \mathcal{I} \to \mathcal{C}$ is a diagram whose limit $\underset{\longleftarrow}{\lim}_{i} X_i$ exists in $\mathcal{C}$, then this limit is preserved by the right adjoint $R$ in that there is a natural isomorphism

   $$R \left( \underset{\longleftarrow}{\lim}_i \left(X_i\right) \right) \;\simeq\; \underset{\longleftarrow}{\lim}_i \left( R(X_i) \right) \,,$$

   where on the right we have the limit in $\mathcal{D}$ over the diagram $R \circ X \colon \mathcal{I} \overset{X}{\longrightarrow} \mathcal{C} \overset{R}{\longrightarrow} \mathcal{D}$.

2. If $X \colon \mathcal{I} \to \mathcal{D}$ is a diagram whose colimit $\underset{\longrightarrow}{\lim}_{i} X_i$ exists in $\mathcal{D}$, then this colimit is preserved by the left adjoint $L$ in that there is a natural isomorphism

   $$L \left( \underset{\longrightarrow}{\lim}_i \left(X_i\right) \right) \;\simeq\; \underset{\longrightarrow}{\lim}_i \left( L(X_i) \right) \,,$$

   where on the right we have the colimit in $\mathcal{C}$ over the diagram $L \circ X \colon \mathcal{I} \overset{X}{\longrightarrow} \mathcal{D} \overset{L}{\longrightarrow} \mathcal{C}$.

Proof

We show the first statement, the proof of the second is formally dual.

We use the following facts

1. There is a natural isomorphism, $Hom_{\mathcal{C}}(L(d),c) \simeq Hom_{\mathcal{D}}(d,R(c))$; this equivalently characterizes the fact that $(L \dashv R)$ is a pair of adjoint functors;

2. (hom-functor preserves limits) The hom-functor sends colimits in the first argument and limits in the second argument to limits of hom-sets

   $$Hom\left( X, \underset{\longleftarrow}{\lim}_i X_i \right) \simeq \underset{\longleftarrow}{\lim}_i Hom\left(X,X_i\right)$$

   and

   $$Hom\left(\underset{\longrightarrow}{\lim}_i X_i, X\right) \simeq \underset{\longleftarrow}{\lim} \left(Hom\left(X_i,X\right) \right) \,.$$

   Again, this is essentially by definition of limits/colimits.

3. (Yoneda lemma) If for two objects $X$ and $Y$ in some category the hom-sets out of or into these objects (their representable functors) are naturally isomorphic, then the two objects are isomorphic, and the isomorphism is obtained by “following the identity” along the natural isomorphisms.

Now using the first two items, we obtain the following chain of natural isomorphisms, for every object $Y \in \mathcal{D}$:

$$\begin{aligned} Hom_{\mathcal{D}}\left( Y, R \left( \underset{\longleftarrow}{\lim}_i X_i \right) \right) & \simeq Hom_{\mathcal{C}}\left( L(Y), \underset{\longleftarrow}{\lim}_i X_i\right) \\ & \simeq \underset{\longleftarrow}{\lim}_i \left( Hom_{\mathcal{C}}\left(L\left(Y\right), X_i\right)\right) \\ & \simeq \underset{\longleftarrow}{\lim}_i \left(Hom_{\mathcal{D}}\left(Y, R\left(X_i\right)\right)\right) \\ & \simeq Hom_{\mathcal{D}}\left(Y, \underset{\longleftarrow}{\lim}_i \left(R\left(X_i\right) \right) \right) \end{aligned} \,.$$

This implies that $R \left( \underset{\longleftarrow}{\lim}_i X_i \right) \cong \underset{\longleftarrow}{\lim}_i \left(R\left(X_i\right) \right)$, but to show $R$ preserves the limit we also need that the isomorphism is given by the “canonical map” $R \left( \underset{\longleftarrow}{\lim}_i X_i \right) \to \underset{\longleftarrow}{\lim}_i \left(R\left(X_i\right) \right)$ given by the universal property. Luckily, we can deduce this simply by following the identity $R \left( \underset{\longleftarrow}{\lim}_i X_i \right) \to R \left( \underset{\longleftarrow}{\lim}_i X_i \right)$ through the chain of natural isomorphisms - naturality of the adjunction implies that the final map we obtain is indeed the canonical map, as required.

Related entries

• hom-functor preserves limits

• limits preserve limits

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• limits and colimits by example

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