202507210207

Tags : Category Theory

Locally Small and Complete Categories with a small coseparating set where all collections of subobjects of a fixed object have an intersection then it is cocomplete

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Theorem

Suppose $C$ is locally small, compelete with a small coseparating set $\Phi$ and has the property that every collection of subobjects of a fixed object has an intersection then $C$ is also cocompelte.

For any small category $J$, if $C$ is locally small, then $C^J$ is also locally small. The constant diagram functor preserves limits. Applying Special Adjunct Functor Theorem we get that $\Delta$ has a left adjoint, which by Limits and Colimits as Adjunctions states that $C$ is cocomplete.

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References

• Special Adjunct Functor Theorem
• Limits and Colimits as Adjunctions
• Complete and Cocomplete Categories
• Small Categories
• Separating and Coseparating sets
• Subobjects