Representable Functors and Limits
- Let $\mathcal{C}$ be a locally small category
- Fix an object $A \in \mathcal{C}$
- For a diagram $D: I \to \mathcal{C}$, we write $\varprojlim D$ for its limit (if it exists) equipped with projections $\pi_i: \varprojlim D \to D(i)$
- For the dual argument, let $\varinjlim D$ denote a colimit equipped with coprojections $\iota_i: D(i) \to \varinjlim D$
- $\operatorname{Hom}_\mathcal{C}(X,Y)$ (or $\mathcal{C}(X,Y)$) denotes the set of morphisms $X \to Y$
Claim. For any object $A$ and any diagram $D: I \to \mathcal{C}$ that admits a limit $\varprojlim D$ we have
\[
\mathcal{C}\big(A, \varprojlim D\big) \cong \varprojlim_{i \in I}\ \mathcal{C}\big(A, D(i)\big)
\]
as sets. Equivalently, the covariant representable functor $\mathcal{C}(A, -): \mathcal{C} \to \mathbf{Set}$ preserves limits.
Proof.
An element of the right-hand side, $\varprojlim \mathcal{C}(A, D(i))$, is defined as a compatible family of morphisms $(f_i: A \to D(i))_{i \in I}$. Compatibility requires that for every arrow $\alpha: i \to j$ in $I$ we have $D(\alpha) \circ f_i = f_j$. This is precisely the definition of a cone from $A$ to the diagram $D$.
Conversely, consider a morphism $f: A \to \varprojlim D$ in the left-hand set. Composing $f$ with the universal projections $\pi_i$ produces the compatible family $(\pi_i \circ f)_{i\in I}$. Thus, we can define a map
\[
\Phi: \mathcal{C}\big(A, \varprojlim D\big) \longrightarrow \varprojlim_{i\in I}\ \mathcal{C}\big(A, D(i)\big) \qquad f \longmapsto (\pi_i \circ f)_{i\in I}
\]
By the universal property of the limit $\varprojlim D$, every compatible family $(f_i)_{i\in I}$ arises uniquely from a single morphism $f: A \to \varprojlim D$ such that $\pi_i \circ f = f_i$ for all $i\in I$. This guarantees both the existence and uniqueness of a preimage for any element in the codomain of $\Phi$. Therefore, $\Phi$ is a bijection.
\(\square\)
Claim (Dual Form). For any object $B$ and any diagram $D: I \to \mathcal{C}$ with colimit $\varinjlim D$,
\[
\mathcal{C}\big(\varinjlim D, B\big) \cong \varprojlim_{i \in I}\ \mathcal{C}\big(D(i), B\big)
\]
Equivalently, the contravariant functor $\mathcal{C}(-, B): \mathcal{C}^{\mathrm{op}} \to \mathbf{Set}$ preserves limits. In ordinary language, $\mathcal{C}(-, B)$ sends colimits in $\mathcal{C}$ to limits in $\mathbf{Set}$.
Proof.
An element of the right-hand side, $\varprojlim \mathcal{C}(D(i), B)$, is a family of morphisms $(g_i: D(i) \to B)_{i\in I}$ satisfying the compatibility condition $g_j \circ D(\alpha) = g_i$ for each arrow $\alpha: i \to j$ in $I$. Such a compatible family is exactly a cocone from the diagram $D$ to $B$.
Given a map $h: \varinjlim D \to B$, precomposing with the universal coprojections $\iota_i: D(i) \to \varinjlim D$ yields the compatible family $(h \circ \iota_i)_{i\in I}$. Thus, we obtain a map
\[
\Psi: \mathcal{C}\big(\varinjlim D, B\big) \longrightarrow \varprojlim_{i\in I}\ \mathcal{C}\big(D(i), B\big), \qquad h \longmapsto (h \circ \iota_i)_{i\in I}
\]
By the universal property of the colimit $\varinjlim D$, every compatible family $(g_i)_{i\in I}$ arises uniquely from a single morphism $h: \varinjlim D \to B$ such that $h \circ \iota_i = g_i$ for all $i\in I$. Hence, $\Psi$ is a bijection.
\(\square\)
While the bijections above are established via universal properties, it is instructive to verify that they are natural in both the object and the diagram. This confirms that representable functors are continuous (limit-preserving).
Limits
Let $D: I \to \mathcal{C}$ be a diagram with limit $\ell = \varprojlim D$ and projections $\pi_i: \ell \to D(i)$. We examine the bijection:
\[
\Phi_{A,D}: \mathcal{C}(A, \ell) \xrightarrow{\cong} \varprojlim_{i \in I}\ \mathcal{C}\big(A, D(i)\big) \qquad \Phi_{A,D}(f) = (\pi_i \circ f)_{i \in I}
\]
We show that $\Phi$ defines a natural isomorphism of functors $\mathcal{C}^{\mathrm{op}} \times [I, \mathcal{C}] \longrightarrow \mathbf{Set}$.
Naturality in the first variable ($A$)
Let $g: A' \to A$ be a morphism in $\mathcal{C}$. We must prove that the following square commutes:
\[
\begin{CD}
\mathcal{C}(A, \ell) @>{\Phi_{A,D}}>> \varprojlim_{i\in I}\ \mathcal{C}(A, D(i)) \\
@V{(-) \circ g}VV @VV{(-) \circ g}V \\
\mathcal{C}(A', \ell) @>{\Phi_{A',D}}>> \varprojlim_{i\in I}\ \mathcal{C}(A', D(i))
\end{CD}
\]
Starting with $f \in \mathcal{C}(A, \ell)$:
- Right-down: First apply $\Phi_{A,D}$ to get $(\pi_i \circ f)_{i\in I}$. Precomposing with $g$ yields the family $(\pi_i \circ f \circ g)_{i \in I}$.
- Down-right: First precompose with $g$ to get $f \circ g$. Applying $\Phi_{A',D}$ yields the family $(\pi_i \circ (f \circ g))_{i \in I}$.
Since composition is associative, $(\pi_i \circ f \circ g) = \pi_i \circ (f \circ g)$ for all $i\in I$. The square commutes, so $\Phi_{-,D}$ is a natural transformation.
Naturality in the diagram ($D$)
Let $\alpha: D \Rightarrow D'$ be a natural transformation between diagrams $D, D': I \to \mathcal{C}$. Let $L(\alpha): \ell \to \ell'$ denote the induced morphism between limits, satisfying $\pi'_i \circ L(\alpha) = \alpha_i \circ \pi_i$ for every $i\in I$. We must show the following square commutes:
\[
\begin{CD}
\mathcal{C}(A, \ell) @>{\Phi_{A,D}}>> \varprojlim_{i\in I}\ \mathcal{C}(A, D(i)) \\
@V{L(\alpha)\circ (-)}VV @VV{\varprojlim\ (\alpha_i \circ -)_{i\in I}}V \\
\mathcal{C}(A, \ell') @>{\Phi_{A,D'}}>> \varprojlim_{i\in I}\ \mathcal{C}(A, D'(i))
\end{CD}
\]
For an element $f: A \to \ell$:
- Down-right: $f$ maps to $L(\alpha)\circ f$. Applying $\Phi_{A,D'}$ gives the family $(\pi'_i\circ L(\alpha)\circ f)_{i\in I}$. Using the property of $L(\alpha)$, this equals $(\alpha_i\circ\pi_i\circ f)_{i\in I}$.
- Right-down: $f$ maps to $(\pi_i \circ f)_{i\in I}$. Applying the induced map on the limit of sets (post-composition with $\alpha_i$) gives $(\alpha_i \circ (\pi_i \circ f))_{i\in I}$.
The results are identical. Thus, $\Phi$ is natural in the diagram argument.
Colimits
Let $D: I \to \mathcal{C}$ have colimit $c = \varinjlim D$ with coprojections $\iota_i: D(i) \to c$ for $i\in I$. We examine the bijection:
\[
\Psi_{D,B}: \mathcal{C}(c, B) \xrightarrow{\cong} \varprojlim_{i \in I}\ \mathcal{C}\big(D(i), B\big) \qquad \Psi_{D,B}(h) = (h \circ \iota_i)_{i \in I}
\]
Naturality in the second variable ($B$)
Let $k: B \to B'$ be a morphism in $\mathcal{C}$. We check the commutativity of:
\[
\begin{CD}
\mathcal{C}(c, B) @>{\Psi_{D,B}}>> \varprojlim_{i\in I}\ \mathcal{C}(D(i), B) \\
@V{k \circ (-)}VV @VV{k \circ (-)}V \\
\mathcal{C}(c, B') @>{\Psi_{D,B'}}>> \varprojlim_{i\in I}\ \mathcal{C}(D(i), B')
\end{CD}
\]
For $h \in \mathcal{C}(c, B)$, following the top-right path gives $(k \circ (h \circ \iota_i))_{i\in I}$. Following the left-bottom path gives $((k \circ h) \circ \iota_i)_{i\in I}$. By associativity, these are equal.
Naturality in the diagram ($D$)
Let $\alpha: D \Rightarrow D'$ be a natural transformation and let $C(\alpha): c \to c'$ be the induced morphism between colimits (satisfying $C(\alpha) \circ \iota_i = \iota'_i \circ \alpha_i$). Consider the diagram below:
\[
\begin{CD}
\mathcal{C}(c', B) @>{\Psi_{D',B}}>> \varprojlim_{i\in I}\ \mathcal{C}(D'(i), B) \\
@V{(-) \circ C(\alpha)}VV @VV{\varprojlim\ (- \circ \alpha_i)_{i\in I}}V \\
\mathcal{C}(c, B) @>{\Psi_{D,B}}>> \varprojlim_{i\in I}\ \mathcal{C}(D(i), B)
\end{CD}
\]
For $h: c' \to B$:
- Left-bottom: $h$ becomes $h \circ C(\alpha)$. Applying $\Psi_{D,B}$ yields $((h \circ C(\alpha)) \circ \iota_i)_{i\in I}$. Using the colimit property, this simplifies to $(h \circ \iota'_i \circ \alpha_i)_{i\in I}$.
- Top-right: $h$ becomes $(h \circ \iota'_i)_{i\in I}$. Precomposing components with $\alpha_i$ yields $((h \circ \iota'_i) \circ \alpha_i)_{i\in I}$.
Equality holds, proving naturality in $D$.
Summary
- The proofs above only use the universal properties of limits and colimits. Consequently, they hold for any diagram shape $I$
- In categorical language, we say that representable functors are continuous. That is, the functor $\mathcal{C}(A, -)$ preserves all limits existing in $\mathcal{C}$
- Dually, the contravariant functor $\mathcal{C}(-, B)$ converts colimits in $\mathcal{C}$ into limits in $\mathbf{Set}$
- This explains why representables are used to detect limits: the limit of a diagram represents the functor that sends an object $X$ to the set of compatible families of maps $(f_i : X \to D(i))_{i\in I}$