nLab hom-functor preserves limits

Idea

One of the basic facts of category theory is that the hom-functor on a category $\mathcal{C}$ preserve limits in both variables separately (remembering that a limit in the first variable, due to contravariance, is actually a colimit in $\mathcal{C}$).

Statement

Ordinary hom-functor

Proposition

(hom-functor preserves limits)

Let $\mathcal{C}$ be a category and write

$$Hom_{\mathcal{C}} \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow Set$$

for its hom-functor. This preserves limits in both its arguments (recalling that a limit in the opposite category $\mathcal{C}^{op}$ is a colimit in $\mathcal{C}$).

More in detail, let $X_\bullet \colon \mathcal{I} \longrightarrow \mathcal{C}$ be a diagram. Then:

1. If the limit $\underset{\longleftarrow}{\lim}_i X_i$ exists in $\mathcal{C}$ then for all $Y \in \mathcal{C}$ there is a natural isomorphism

   $$Hom_{\mathcal{C}}\left(Y, \underset{\longleftarrow}{\lim}_i X_i \right) \simeq \underset{\longleftarrow}{\lim}_i \left( Hom_{\mathcal{C}}\left( Y, X_i \right) \right) \,,$$

   where on the right we have the limit over the diagram of hom-sets given by

   $$Hom_{\mathcal{C}}(Y,-) \circ X \;\colon\; \mathcal{I} \overset{X}{\longrightarrow} \mathcal{C} \overset{Hom_{\mathcal{C}}(Y,-) }{\longrightarrow} Set\,.$$

2. If the colimit $\underset{\longrightarrow}{\lim}_i X_i$ exists in $\mathcal{C}$ then for all $Y \in \mathcal{C}$ there is a natural isomorphism

   $$Hom_{\mathcal{C}}\left(\underset{\longrightarrow}{\lim}_i X_i ,Y\right) \simeq \underset{\longleftarrow}{\lim}_i \left( Hom_{\mathcal{C}}\left( X_i , Y\right) \right) \,,$$

   where on the right we have the limit over the diagram of hom-sets given by

   $$Hom_{\mathcal{C}}(-,Y) \circ X \;\colon\; \mathcal{I}^{op} \overset{X}{\longrightarrow} \mathcal{C}^{op} \overset{Hom_{\mathcal{C}}(-,Y) }{\longrightarrow} Set\,.$$

Proof

We give the proof of the first statement. The proof of the second statement is formally dual.

First observe that, by the very definition of limiting cones, maps out of some $Y$ into them are in natural bijection with the set $Cones\left(Y, X_\bullet \right)$ of cones over the diagram $X_\bullet$ with tip $Y$:

$$Hom\left( Y, \underset{\longleftarrow}{\lim}_{i} X_i \right) \;\simeq\; Cones\left( Y, X_\bullet \right) \,.$$

Hence it remains to show that there is also a natural bijection like so:

$$Cones\left( Y, X_\bullet \right) \;\simeq\; \underset{\longleftarrow}{\lim}_{i} \left( Hom(Y,X_i) \right) \,.$$

Now, again by the very definition of limiting cones, a single element in the limit on the right is equivalently a cone of the form

$$\left\{ \array{ && \ast \\ & {}^{\mathllap{const_{p_i}}}\swarrow && \searrow^{\mathrlap{const_{p_j}}} \\ Hom(Y,X_i) && \underset{X_\alpha \circ (-)}{\longrightarrow} && Hom(Y,X_j) } \right\}_{i, j \in Obj(\mathcal{I}), \alpha \in Hom_{\mathcal{I}}(i,j) } \,.$$

This is equivalently for each object $i \in \mathcal{I}$ a choice of morphism $p_i \colon Y \to X_i$ , such that for each pair of objects $i,j \in \mathcal{I}$ and each $\alpha \in Hom_{\mathcal{I}}(i,j)$ we have $X_\alpha \circ p_i = p_j$. And indeed, this is precisely the characterization of an element in the set $Cones\left( Y, X_{\bullet} \right)$.

Internal hom-functor

Proposition

(internal hom-functor preserves limits)

Let $\mathcal{C}$ be a symmetric closed monoidal category with internal hom-bifunctor $[-,-]$. Then this bifunctor preserves limits in the second variable, and sends colimits in the first variable to limits:

$$[X, \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} Y(j)] \;\simeq\; \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [X, Y(j)]$$

and

$$[\underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j),X] \;\simeq\; \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [Y(j),X]$$

Proof

For $X \in \mathcal{C}$ any object, $[X,-]$ is a right adjoint by definition, and hence preserves limits as adjoints preserve (co-)limits.

For the other case, let $Y \;\colon\; \mathcal{L} \to \mathcal{C}$ be a diagram in $\mathcal{C}$, and let $C \in \mathcal{C}$ be any object. Then there are isomorphisms

$$\begin{aligned} Hom_{\mathcal{C}}(C, [ \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j), X ] ) & \simeq Hom_{\mathcal{C}}( C \otimes \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j), X ) \\ & \simeq Hom_{\mathcal{C}}( \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} (C \otimes Y(j)), X ) \\ & \simeq \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} Hom_{\mathcal{C}}( (C \otimes Y(j)), X ) \\ & \simeq \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} Hom_{\mathcal{C}}( C, [Y(j), X] ) \\ & \simeq Hom_{\mathcal{C}}( C, \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [Y(j), X] ) \end{aligned}$$

which are natural in $C \in \mathcal{C}$, where we used that the ordinary hom-functor respects (co)limits as shown (see at hom-functor preserves limits), and that the left adjoint $C \otimes (-)$ preserves colimits (see at adjoints preserve (co-)limits).

Hence by the fully faithfulness of the Yoneda embedding, there is an isomorphism

$$\left[ \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j), X \right] \overset{\simeq}{\longrightarrow} \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [Y(j), X] \,.$$

Related statements

• limits preserve limits

• adjoints preserve (co-)limits

• limits commute with limits

• limits of presheaves are computed objectwise