Weakly Initial Family

A weakly initial family in a Category $\mathcal{C}$ is a family of objects $X_{i}:\mathcal{C}$ such that for every $Y:\mathcal{C}$, there Merely exists a morphism $f:\mathcal{C}(X_{i},Y)$ for some $i:I$.

Properties

• When $\mathcal{C}$ is $\kappa$-Complete, we can upgrade a $\kappa$-small weak initial family into an Initial Object.

  The idea is to take an Indexed Cartesian Product $\prod_{(i:I)}X_{i}$ over the weak initial family, which allows us to upgrade our weakly initial family into a weak initial object by constructing a (not neccesarily unique) map $\prod_{(i:I)}X_{i}\to Y$ by using the fact that there must be an $i:I$ and map $h:X_{i}\to Y$, which we can precompose with the $i$th projection $\pi_{i}$.

  Moreover, we can then take a Joint Equaliser $E$ over all the endomaps $\prod_{(i:I)}\to\prod_{(i:I)}$; which upgrades the weak initial object to an Initial Object.