(Co)Limit (Category Theory)

Idea

In category theory, limits and colimits express a certain kind of universal property that identifies a universal element, called the apex paired with morphisms to all objects in the diagram (a cone, or dually a cocone) such that all other cones/cocones factor uniquely through it. These are a general tool used to express pullbacks/pushouts, products/coproducts. They can generalize to multi-(co)limits enbling expression of end and coend.

Limit

In category theory, a limit of a diagram $D:\mathcal{I}\Rightarrow \mathcal{C}$ is the terminal object of the comma category $(\Delta \downarrow D)$, where $\Delta :\mathcal{C}\Rightarrow [\mathcal{I},\mathcal{C}]$ is the functor sending $d$ to the constant diagram $\Delta d$. We regard $D$ in $(\Delta \downarrow D)$ as the constant functor from the terminal category $1$ to $[\mathcal{I},\mathcal{C}]$, picking out the diagram $D$.

The objects of $(\Delta \downarrow D)$ come in the form $(d:\mathcal{C},\mathcal{N}:\Delta d\Rightarrow D)$. The terminal such object is an object $d$ (the apex) and a natural transformation $\mathcal{N}$ (the cone). Naturality ensures that for all $f:a\to b$ in $D$, the triangle¹ $f\circ {\alpha }_a={\alpha }_b$ commutes:

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
d \arrow[d, "\alpha_a"] \arrow[r, "id_d"] & d \arrow[d, "\alpha_b"] \\
a \arrow[r, "f"] & b
\end{tikzcd}
\end{document}

Colimit

In category theory, a colimit of a diagram $D:\mathcal{I}\Rightarrow \mathcal{C}$ is the initial object of the comma category $(D\downarrow \Delta )$, where $\Delta :\mathcal{C}\Rightarrow [\mathcal{I},\mathcal{C}]$ is the constant diagram functor. We regard $D$ as before.

The objects of $(D\downarrow \Delta )$ come in the form $(d:\mathcal{C},\mathcal{N}:D\Rightarrow \Delta d)$. The initial such object is the colimit: $d$ is the apex and $\mathcal{N}$ the cocone. Naturality ensures that for all $f:a\to b$ in $D$, the triangle¹ ${\alpha }_b\circ f={\alpha }_a$ commutes:

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
a \arrow[d, "\alpha_a"] \arrow[r, "f"] & b \arrow[d, "\alpha_b"] \\
d \arrow[r, "id_d"] & d
\end{tikzcd}
\end{document}

Remarks

• The limit of an opposite diagram is a colimit, and vice versa.
• A limit/colimit is finite iff the index category is finite.
• Limits and colimits can be expressed via adjunctions.
• By the Yoneda Lemma, limits represent certain functors.
• The colimit of an Opposite Category is a limit of the opposite diagram.

Examples

• The product of two objects is a limit of a diagram with two objects and no nontrivial morphisms.
• A pushout is a colimit.
• A pullback is a limit.

Footnotes

1. Drawn as a square due to technical limitations. ↩ ↩²