Limits & Colimits
Terminal objects, products, equalizers and pullbacks are one construction read over four different shapes.
The idea
Limits and colimits are the general form of the universal constructions built so far. Terminal objects, products, equalizers and pullbacks were defined by the same demand — for every object carrying such-and-such arrows, exactly one arrow to ours makes everything commute — and they differ only in the shape of the data the demand starts from. The general definition makes the shape an input.
Let $\mathsf{I}$ be a small category, one with only a set's worth of arrows. A diagram of shape $\mathsf{I}$ in a category $\mathcal{A}$ is a functor $D \colon \mathsf{I} \to \mathcal{A}$, so $\mathsf{I}$ contributes no mathematics of its own, only a pattern of objects and arrows to be filled in.
Definition (Limit).
A cone on $D$ is an object $A$, its vertex, together with an arrow $f_{I} \colon A \to D(I)$ for each object $I$ of $\mathsf{I}$, such that $D(u) \circ f_{I} = f_{J}$ for every arrow $u \colon I \to J$. A limit of $D$ is a universal cone $(L, (p_{I}))$: every cone factors through it by exactly one arrow compatible with all the $p_{I}$.
Reversing every arrow turns cones into cocones and limits into colimits. Four shapes recover everything already built, together with the duals: the empty shape gives terminal and initial objects, a shape with no arrows but identities gives products and coproducts, a parallel pair gives equalizers and coequalizers, and a corner gives pullbacks and pushouts. One definition now covers eight constructions.
Ways to work on it
- Walkthrough. Diagrams as functors, cones, the limit as a universal cone, and the four familiar constructions read off four shapes.
- Proof. Products and equalizers build every small limit: one product over the objects, one over the arrows, and an equalizer between them.
- Practice. Compute limits and colimits over assorted small shapes, and classify which is which.
- Hardest. Push the limit and colimit constructions past the familiar examples.
Not sure where to start? Take the ten-question placement test.