General Limits

Summary

A limit of a diagram $D:\mathcal{I}\to \mathcal{A}$ is a terminal cone: an object $\lim D$ with a natural transformation $\lambda :\Delta (\lim D)\Rightarrow D$ through which every cone factors uniquely. Limits in Set have a concrete formula as a subset of a product. Products + equalizers generate all limits.

Overview

The general definition of limit unifies products, equalizers, pullbacks, and all other “meet-like” constructions. A diagram is just a functor $D:\mathcal{I}\to \mathcal{A}$, a cone is a compatible family of maps into the diagram, and the limit is the universal cone.

Main Content

Definition 5.1.18: Diagram

A diagram in $\mathcal{A}$ of shape $\mathcal{I}$ is a functor $D:\mathcal{I}\to \mathcal{A}$. The category $\mathcal{I}$ is called the index category (or shape category).

Common index categories:

• $\mathcal{I}=\bullet$: a single object, no non-identity maps. Limit = the object itself.
• $\mathcal{I}=\bullet \bullet$: two objects, no maps between them. Limit = product.
• $\mathcal{I}=\bullet \rightrightarrows \bullet$: parallel pair. Limit = equalizer.
• $\mathcal{I}=\bullet \to \bullet \leftarrow \bullet$: cospan. Limit = pullback.
• $\mathcal{I}=\varnothing$: empty diagram. Limit = terminal object.

Definition 5.1.19: Cone

A cone over a diagram $D:\mathcal{I}\to \mathcal{A}$ with vertex $A\in \mathcal{A}$ is a natural transformation $\lambda :\Delta A\Rightarrow D$, where $\Delta A:\mathcal{I}\to \mathcal{A}$ is the constant functor at $A$.

Concretely, a cone consists of:

• Maps ${\lambda }_I:A\to DI$ for each $I\in \mathcal{I}$
• Such that for every $u:I\to J$ in $\mathcal{I}$: $Du\circ {\lambda }_I={\lambda }_J$

Definition 5.1.20: Limit

A limit of $D:\mathcal{I}\to \mathcal{A}$ is a terminal cone: a cone $\lambda :\Delta L\Rightarrow D$ such that for every cone $\mu :\Delta A\Rightarrow D$, there is a unique map $\bar{\mu }:A\to L$ with ${\lambda }_I\circ \bar{\mu }={\mu }_I$ for all $I\in \mathcal{I}$.

Write $L=\lim D$ or ${\lim }_{\mathcal{I}}D$ or ${\lim }_{I\in \mathcal{I}}DI$.

Limits are unique up to isomorphism when they exist (by terminality).

Limits in Set

Example 5.1.22: Limit Formula in Set (BCT, Ch. 5.1)

For a diagram $D:\mathcal{I}\to \mathbf{Set}$, the limit is:

$$\lim D=\left\{(x_I{)}_{I\in \mathcal{I}}\in \prod _{I\in \mathcal{I}}DI|\forall u:I\to J\text{ in }\mathcal{I},(Du)(x_I)=x_J\right\}$$

with projections ${\lambda }_I:\lim D\to DI$, $(x_I{)}_I\mapsto x_I$.

I.e., the limit is the set of compatible families of elements.

Example: Special Cases of the Set Limit Formula

• Product: $\lim (A_i{)}_{i\in I}={\prod }_i{A}_i$ (no compatibility conditions when $\mathcal{I}$ has no non-identity maps).
• Equalizer: $\lim (s,t:X\rightrightarrows Y)=\{x\in X:s(x)=t(x)\}$.
• Pullback: $X{\times }_{Z}Y=\{(x,y):g(x)=f(y)\}$.
• Terminal object: $\lim (\varnothing )=\{\ast \}$.

Completeness

Definition: Complete Category

A category $\mathcal{A}$ is complete if it has all small limits (i.e., limits of all diagrams $D:\mathcal{I}\to \mathcal{A}$ with $\mathcal{I}$ small).

$\mathcal{A}$ is finitely complete if it has all finite limits.

Examples: Set, Top, Grp, Ab, Ring, $[\mathcal{A},\mathcal{B}]$ (when $\mathcal{B}$ is complete) are all complete.

Products + Equalizers Generate All Limits

Proposition 5.1.26 (BCT, Ch. 5.1)

A category has all small limits if and only if it has all small products and all equalizers.

For finite limits, the same with “finite.”

Connections

• Colimits (Colimits) are limits in the opposite category: initial cones.
• Functors and limits (Functors and Limits): not all functors preserve limits; continuous functors are those that do.
• Limits via representables (Limits via Representables): $\lim D$ represents the functor $\mathrm{Cone}(-,D)$.
• Right adjoints preserve limits (Adjoints and Limits).

See Also

• Products and Equalizers — Special cases
• Pullbacks — Another special case
• Colimits — Dual notion
• Functors and Limits — Preservation and reflection
• Limits via Representables — Limits as representable functors