202506020206

Tags : Category Theory

Representable Universal Property of Colimits

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The Idea is very similar to Representable Universal Property of Limits, and the plan is to use duality. For a functor $F:J\to C$ and $X:C$ we can define

$$C(F-,X):\equiv J^{\text{op}}\stackrel{F}{\to }{C}^{\text{op}}\stackrel{C(-,X)}{\to }\text{Set}$$

Again, we know that the limit exists and can be constructed as an equalizer. Hence elements of ${\text{lim}}_{J}C(F-,X)$ are elements of ${\prod }_{j\in J}C(Fj,X)$, such that the following diagram commutes thus we have ${\text{lim}}_{J}C(F-,X)$ thus we have the theorem

Theorem

For any diagram $J$ whose colimit exists, there is a natural isomorphism

$$C({\text{colim}}_{J}F,X)\simeq {\text{lim}}_{J^{\text{op}}}C(F-,X)$$

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References

• Representable Universal Property of Limits
• Examples of Representable Universal Property of Colimits