Notes on Adjoints, Representables, Limits

18 Oct 2026 PDF

Functorialness of Cones

Define the diagonal functor $\Delta : \mathcal{A} \to [ \mathbf{I}, \mathcal{A} ]$, where $\Delta (A) : \mathbf{I} \to \mathcal{A}$ maps every element and map of $\mathbf{I}$ to $A$ and $1_A$, respectively.

We then have,

\[\mathrm{Cone}(A, D) = \hom_{[\mathbf{I}, \mathcal{A}]}(\Delta A, D)\]

Hence, $\mathrm{Cone}(A, D)$ is functorial in $A$ (contravariantly) and $D$ (covariantly).

Limits as Representables

Varying $A$ yields the following theorem:

A limit of $D$ is a representation of $\hom_{[\mathbf{I}, \mathcal{A}]}(\Delta -, D)$.

Varying $D$ yields the following theorem:

Let $\alpha$ be a natural transformation between $D, D’ : \mathbf{I} \to \mathcal{A}$. The limits of $D, D’$ induce $\lim_{\leftarrow I}\alpha : \lim_{\leftarrow I} D \to \lim_{\leftarrow I}D’$ such that,

• (a) commutes
• (c) commutes if (b) commutes

Limits as Adjoints

If $\mathcal{A}$ has all limits of shape $\mathbf{I}$, we can give a second reformulation:

$\lim_{\leftarrow \mathbf{I}} : [\mathbf{I}, \mathcal{A}] \to \mathcal{A}$ is the right adjoint to the diagonal functor $\Delta : \mathcal{A} \to [ \mathbf{I}, \mathcal{A} ]$.

Representables preserve Limits

For a locally small category $\mathcal{S}$ that has products and equalizers, one can construct limits for arbitrary $D: \mathbf{I} \to \mathcal{S}$ by taking the equalizer of the following:

\[\prod_{I \in \mathbf{I}} D(I)\; \substack{\underrightarrow{\;\;s\;\;}\\ \overrightarrow{\;\;t\;\;}} \prod_{I \xrightarrow{u} J} D(J)\]

where $s_u = Du \circ \pi_I$ and $t_u = \pi_J$. In light of this, we have:

\[\mathrm{Cone}(A, D) \cong \lim_{\leftarrow I}\hom_\mathcal{A} (A, D(-))\]

Proof. Since $\hom_\mathcal{A}(A, D(-))$ is a functor into $\mathbf{Set}$, its limit is given by the above formula, which is the set of all families $\big(f_I : A \to D(I)\big)_{I \in \mathbf{I}}$ such that $Du \circ f_I = f_J$, which happens to coincide with $\mathrm{Cone}(A, D)$. ■

Since $\mathrm{Cone}(A, D) \cong \hom_\mathcal{A} (A, \lim D)$, we have

\[\hom_\mathcal{A}\left(A, \lim_{\leftarrow \mathbf{I}} D\right) \cong \lim_{\leftarrow \mathbf{I}} \hom_\mathcal{A} (A, D(-))\]

that is, representables preserve limits.

To dualize the result, note that $\hom_\mathcal{A}(-, A)$ is a contravariant functor from $\mathcal{A}$, and hence a covariant functor from $\mathcal{A}^{\mathrm{op}}$. Hence, $\hom_\mathcal{A}(-, A) : \mathcal{A}^{\mathrm{op}} \to \mathbf{Set}$ preserves limits in $\mathcal{A}^\mathrm{op}$, i.e.

\[\hom_\mathcal{A}\left( \lim_{\leftarrow \mathbf{I}^{\rm{op}}} D^{\rm{op}}, A\right) \cong \lim_{\leftarrow \mathbf{I}^{\rm{op}}} \hom_\mathcal{A} (D^{\rm{op}}(-), A)\]

where $D^{\rm{op}}: \mathbf{I}^{\rm{op}} \to \mathcal{A}^{\rm{op}}$. If we dualize $D^\mathrm{op}$ to $D: \mathbf{I} \to \mathcal{A}$, we have $\lim_{\leftarrow \mathbf{I}^{\rm{op}}} D^{\rm{op}} = \lim_{\rightarrow \mathbf{I}} D$.

Hence,

\[\hom_\mathcal{A}\left( \lim_{\rightarrow \mathbf{I}} D, A\right) \cong \lim_{\leftarrow \mathbf{I}} \hom_\mathcal{A} (D(-), A).\]

Note that RHS is limit, not colimit.

Corollary. Left adjoints preserve colimits and right adjoints preserve limits.

Proof. From the definition of adjointness and $\mathrm{Cone}(A, D) \cong \lim \hom_\mathcal{A} (A, D(-)) \cong \hom_\mathcal{A}(A, \lim D)$, we have

\[\begin{align} \hom_\mathcal{A} (A, G(\lim D)) &\cong \hom_\mathcal{B} (F(A), \lim D) \\ &\cong \lim \hom_\mathcal{B} (F(A), D(-)) \\ &\cong \lim \hom_\mathcal{A}(A, GD(-)) \\ &\cong \mathrm{Cone}(A, G\circ D). \end{align}\]

Hence, $G(\lim D)$ represents $\mathrm{Cone}(-, G \circ D)$, i.e. $G(\lim D) \cong \lim G \circ D$. ■

Limits in Functor Categories are Pointwise-computed

Theorem. Let $D: \mathbf{I} \to [\mathbf{A}, \mathcal{S}]$ be a diagram, and that for each $A \in \mathbf{A}$, $D(-)(A) : \mathbf{I} \to \mathcal{S}$ has a limit. Then there exists a cone on $D$ such that its image under $\mathrm{ev}_A$ is a limit cone on $D(-)(A)$ for each $A \in \mathbf{A}$. Moreover, any such cone on $D$ is a limit cone.

Proof. Follows from “Limits as Representables, varying $D$”. Refer to the following diagram.

Corollaries.

1. Limits commute with limits.
2. $\mathrm{ev}_A : [\mathbf{A} , \mathcal{S} ] \to \mathcal{S}$ preserves limits.
3. $よ: \mathbf{A} \to [\mathbf{A}^{\rm{op}}, \mathbf{Set}]$ preserves limits.

Density Theorem

Theorem. Every presheaf is a colimit of representables. More precisely, for $X : \mathbf{A}^{\rm{op}} \to \mathbf{Set}$, $X$ is the colimit of

\[\mathbf{E}(X) \xrightarrow{P} \mathbf{A} \xrightarrow{よ} [\mathbf{A}^{\rm{op}}, \mathbf{Set}].\]