I read about what an infimum is and it makes sense, but there is something about this notation that is confusing to me.
It is from this paper
Recall that in regression, the forecaster $H$ outputs at each step $t$ a CDF $F_t$ targeting $y_t$. We will use $F_t^{-1}:[0,1] \rightarrow \mathcal{Y}$ to denote the quantile function $F_t^{-1}(p) = \inf\{y: p \le F_t(y)\}$
If $y$ is the value of the CDF, shouldn't the quantile function be the supremum y value that is less than or equal to $F_t(y)$?
As $F_t$ is a CDF, it is an increasing function. That means that larger $y$ gives larger $F_t(y)$. Optimally, we want $F^{-1}_t(p)$ to be the smallest $y$ such that $F_t(y)=p$. So that's what it's defined to be. All the other notation is just there to make sure that the value actually exists and is well-defined. If $F_t$ is known to be continuous, it is unnecessary, but if $F_t$ is allowed to be discontinuous, all that extra stuff is necessary.
By the increasing nature of $F_t$, we may as well look for the smallest $y$ such that $F_t(y)\geq p$, except that $y$-values such that $F_t(y)\geq p$ are certain to exist. Then we pick out the smallest such $y$, except there might not be a smallest, so we take the $\inf$ instead.