Generalized Tukey reducibility between $\sigma$-directed sets

We introduce the pre-Tukey reducibility, a generalization of the Tukey reducibility between directed sets that works well in $\mathsf{ZF}$. We investigate the pre-Tukey reducibility between several $\sigma$-directed sets under assumptions on sets of reals, which hold in the Solovay model and in $L(\mathbb{R})$ satisfying $\mathsf{AD}$.