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 $\mathcal{C}$).

In more 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 $Y \in \mathcal{C}$ the functor $Hom_{\mathcal{C}}(Y, -)$ preserves this limit.

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

2. If the colimit $\underset{\longrightarrow}{\lim}_i X_i$ exists in $\mathcal{C}$ then for all $Y \in \mathcal{C}$ the functor $Hom_{\mathcal{C}}(-, Y)$ preserves it (viewing it as a limit over $X_\bullet^{op}$).

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

Proof

We give the proof of the first statement. The proof of the second statement is formally dual. Let $(L, \lambda_i)$ be a limit of $X_\bullet$, consisting of an object $L \in \mathcal{C}$ and a family of maps $\lambda_i \colon L \to X_i$ forming a limit cone. Our task is to show that $(Hom_\mathcal{C}(Y, L), Hom_\mathcal{C}(Y, \lambda_i))$ is a limit cone in $Set$.

So, take a set $S$ equipped with a family of maps $\alpha_i \colon S \to Hom_\mathcal{C}(Y, X_i)$ that form a cone, meaning for every $f \colon X_i \to X_j$ we have that $Hom_\mathcal{C}(Y, f) \circ \alpha_i = \alpha_j$. We need to show that there exists a unique morphism $\alpha \colon S \to Hom_\mathcal{C}(Y, L)$ such that $\alpha_i = Hom_\mathcal{C}(Y, \lambda_i) \circ \alpha$ for every $i \in \mathcal{I}$.

Unpacking the definition of the hom-functor, we obtain:

• For every $s \in S$, a map $\alpha_i(s) \colon Y \to X_i$.
• The compatibility condition $f \circ \alpha_i(s) = \alpha_j(s)$ for every $f \colon X_i \to X_j$.
• The requirement to find, for each $s \in S$, a unique $\alpha(s) \colon Y \to L$ such that $\alpha_i(s) = \lambda_i \circ \alpha(s)$.

But elementwise, the first two bulletpoints are the data of a cone over $X_\bullet$ with tip $Y$, and the third is obtained from the unique morphism $Y \to L$ factorising this cone through the limit cone $(L, \lambda_i)$. Thus, taking these together for each $s \in S$ provides the required unique factorisation $\alpha \colon S \to Hom_\mathcal{C}(Y, L)$.

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