# Evaluation of limit and colimit in the category of Set

Given two objects $A,B$ in Set with an arrow $A\xrightarrow{f} B$ between them, find the limit and colimit of this diagram.

By definition, the limit is the unique universal left solution, that is, it is an object $X$ in Set with arrows $X\xrightarrow{s_A} A$ and $X\xrightarrow{s_B}B$ making the appropriate diagram commute (I'd draw it but I'm not sure how). We must also have that for any other left solution $S$, there is a unique arrow $X\to S$, again making the correct diagrams commute.

With this in mind, I think that the limit of the diagram should be the null set, since there is certainly a unique map from the null set into any other possible solution, and commutivity would be trivially satisfied. However, I'm not really sure what the colimit should be. I think it might be the power set of $A\cup B$, but I'm not really sure how to show this. Any suggestions?

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Your definition of "limit" is wrong. – Chris Eagle Apr 26 '12 at 19:39
If you feel like generalizing this, you should prove that if a diagram has a terminal object $B$, then then the colimit of the diagram is $B$. Similarly, if your diagram has an initial object $A$, then the limit of the diagram is $A$. – SL2 Apr 26 '12 at 20:02

The limit should receive an arrow for any possible solution to the problem, so the limit should just be $A$ with the identity map to itself. The colimit is similar.
Okay. After referring back to my text, I realized I copied the arrows pointing in the wrong direction. The limit should be able to receive any solution, and the colimit should have an arrow to any possible solution. Thus, the colimit should be the object $B$ with an identity arrow. Thank you. – chris Apr 26 '12 at 20:24