202507191607

Tags : Category Theory

Forgetful functor from comma category strictly creates limits

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Theorem

For any functor $U:A\to S$ and object $s$, the associated forgetful functor $\prod :s\downarrow U\to A$ strictly creates the limit of any diagram whose limits exist in $A$ and is preserved by $U$. In particular, if $A$ is complete and $U$ is continuous, then $s\downarrow U$ is complete.

The proof is an extension to Forgetful functor from comma category strictly creates limits.

This seem to imply that all continuous functors from complete categories should admit left adjoints. This is not the case because $s\downarrow U$ is not necessarily small, so even if $A$ admits all small limits, it may not admit limits of large diagrams.

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References

• Complete and Cocomplete Categories
• Continuous and Cocontinuous Functor
• Forgetful functor from comma category strictly creates limits