202506091506

Tags : Category Theory

Set is Complete and Cocomplete

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We have already shown that Set is Complete. From Any Category with Coproducts and Coequalizers is Cocomplete, with Products and Equalizers is Complete we will get that $\text{Set}$ is also co-complete.

To do so we need to show that $\text{Set}$ contains all Coproducts and Coequalizers.

• Coproducts in the category of sets are disjoint unions
• For coequalizers, given functions $f,g:A\to B$, the coequalizer is the set $C$ which is given by $C=B/(f=g)$ that is the set $B$ with elements $a,b$ quotiented together if there is some $x:A$ such that $f(x)=a$ and $g(x)=b$.

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References

• Any Category with Coproducts and Coequalizers is Cocomplete, with Products and Equalizers is Complete
• Top is Complete and Cocomplete
• Complete and Cocomplete Categories