Set Category

up:: Category

The category $Set$ is the category whose objects are sets $X$ and whose morphisms are functions $X\stackrel{f}{\to }Y$.

Properties

• Note that, since for any set $X,Y$, the cardinality of the set of functions from $X$ to $Y$ is $|Y{|}^{|X|}$, we have that $Set$ is Locally Small.
• It is not a Small Category, since this would imply that the set of all sets is itself a set (which would violate Russell’s Paradox)

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References

• Set in nLab