Franklin's Notes

Complete category

A category $\mathsf C$ is called complete if it has all small limits , and similarly cocomplete if it has all small colimits. A category is called finitely complete if every finite diagram (i.e. with finitely many objects and arrows) has a limit, and finitely cocomplete if every finite diagram has a colimit.

Theorem 1. If $\mathsf C$ has a terminal object and a pullback for each pair of arrows with a common codomain, then it is finitely complete.

I'm still working on understanding why this theorem is true, and I'll add a proof whenever I accomplish this.



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