What are the uses of Limits and Colimits of Category Theory in every day problems? I am interested in knowing how we can use the concepts of Limits and Colimits in modeling problems in every day life? Could anyone provide (Software) engineering examples, perhaps? Or describe intuitively in general for what sorts of modeling problems we can use these concepts? Thank you.
Take some relations $R_0\subseteq A_0\times A_1$, $R_1\subseteq A_1\times A_2$. Let $\pi_1:R_0\to A_1$, $\pi_0:R_1\to A_1$ be projections with domains restricted to relations $R_0$, $R_1$. Then the pullback of $\pi_0$, $\pi_1$ is the join of $R_0$ and $R_1$ in the SQL sense.