Limits via Representables

Summary

The cone functor $\mathrm{Cone}(-,D):{\mathcal{A}}^{\mathrm{op}}\to \mathbf{Set}$ sending $A\mapsto \mathrm{Cone}(A,D)$ is a presheaf. The limit $\lim D$ is exactly the representing object of this functor: $\mathrm{Cone}(A,D)\cong \mathcal{A}(A,\lim D)$ naturally. Consequently, limits are unique up to isomorphism, and the limit functor $\lim :[\mathcal{I},\mathcal{A}]\to \mathcal{A}$ is right adjoint to the diagonal functor $\Delta$.

Overview

Chapter 6 unifies the three main themes of the book. It begins by showing that limits are representable functors (Ch. 6.1). This is the formal unification: the language of representability and the Yoneda lemma now governs limits directly.

Main Content

Cones as Natural Transformations

Recall that a cone over $D:\mathcal{I}\to \mathcal{A}$ with vertex $A$ is a natural transformation $\Delta A\Rightarrow D$ in $[\mathcal{I},\mathcal{A}]$.

Proposition 6.1.1: Cones are Representable (BCT, Ch. 6.1)

There is a natural isomorphism

$$\mathrm{Cone}(A,D)\cong [\mathcal{I},\mathcal{A}](\Delta A,D)$$

So the functor $\mathrm{Cone}(-,D):{\mathcal{A}}^{\mathrm{op}}\to \mathbf{Set}$, $A\mapsto \mathrm{Cone}(A,D)$, is the functor $[\mathcal{I},\mathcal{A}](\Delta -,D)$.

A limit of $D$ is a representation of $\mathrm{Cone}(-,D)$: an object $L\in \mathcal{A}$ and a natural isomorphism $\mathrm{Cone}(-,D)\cong \mathcal{A}(-,L)$.

Limits are Unique Up to Isomorphism

Corollary 6.1.2: Limits Are Unique (BCT, Ch. 6.1)

If $L$ and $L^{\prime }$ both represent $\mathrm{Cone}(-,D)$ (i.e., are both limits of $D$), then $L\cong L^{\prime }$ via a unique isomorphism compatible with the universal cones.

Proof: Any two representations of the same functor are isomorphic, by the Yoneda lemma (^unique-rep).

The Limit Functor and Its Adjoint

Proposition 6.1.4: $\lim ⊣\Delta$ (BCT, Ch. 6.1)

If $\mathcal{A}$ has all limits of shape $\mathcal{I}$, then the limit construction is a functor $\lim :[\mathcal{I},\mathcal{A}]\to \mathcal{A}$ that is right adjoint to the diagonal functor $\Delta :\mathcal{A}\to [\mathcal{I},\mathcal{A}]$.

The adjunction bijection is:

$$[\mathcal{I},\mathcal{A}](D,\Delta A)\cong \mathcal{A}(A,\lim D)$$

Wait — actually this is for limits: $[\mathcal{I},\mathcal{A}](\Delta A,D)\cong \mathcal{A}(A,\lim D)$, so $\Delta ⊣\lim$.

Correction: It is $\Delta ⊣\lim$ (diagonal is left adjoint to limit). The unit is the universal cone ${\eta }_D:\Delta (\lim D)\Rightarrow D$; the counit is ${\epsilon }_A:\lim (\Delta A)\stackrel{\sim }{\to }A$ (limit of constant diagram is that object).

Dually, $\mathrm{colim}⊣\Delta$: the diagonal is right adjoint to the colimit functor.

Limits of Functors

Example: Limits of Sequences (BCT, Ch. 6.1)

For $\mathcal{I}={\mathbb{N}}^{\mathrm{op}}$ (reverse natural numbers as a poset), a diagram $D:\mathcal{I}\to \mathcal{A}$ is a sequence $D_0\leftarrow D_1\leftarrow D_2\leftarrow \cdots$ (inverse system). Its limit is the inverse limit $\underleftarrow{\lim }{D}_n$.

Example in Set: $\underleftarrow{\lim }(\mathbb{Z}/p^n\mathbb{Z})={\mathbb{Z}}_p$ (the $p$-adic integers).

Example: Limits Commute with Limits (BCT, Prop. 6.2.8)

For any diagram shape $\mathcal{I}$ and $\mathcal{J}$ and any diagram $D:\mathcal{I}\times \mathcal{J}\to \mathcal{A}$:

$$\underset{I\in \mathcal{I}}{\lim }\underset{J\in \mathcal{J}}{\lim }D(I,J)\cong \underset{J\in \mathcal{J}}{\lim }\underset{I\in \mathcal{I}}{\lim }D(I,J)\cong \underset{\mathcal{I}\times \mathcal{J}}{\lim }D$$

whenever the relevant limits exist.

Connections

• Limits in presheaf categories (Limits in Presheaf Categories): in $[{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}]$, limits are pointwise — this is computed via the representability of cones.
• Adjoints and limits (Adjoints and Limits): the right adjoint $\lim$ preserves limits; the left adjoint $\mathrm{colim}$ preserves colimits.
• Yoneda (Yoneda Embedding and Consequences): the Yoneda embedding preserves limits, proven using this representability.

See Also

• General Limits — Limit definition
• Representable Functors — What representability means
• Yoneda Lemma — Used to prove uniqueness
• Adjoints and Limits — Adjoint functors and limits