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We study the lattice structure of the family of weakly compact subsets of theunit ball $B_X$ of a separable Banach space $X$, equipped with the inclusionrelation (this structure is denoted by $\mathcal{K}(B_X)$) and also with theparametrized family of almost inclusion relations $K \subseteq L+\epsilon B_X$,where $\epsilon>0$ (this structure is denoted by $\mathcal{AK}(B_X)$). Tukeyequivalence between partially ordered sets and a suitable extension to dealwith $\mathcal{AK}(B_X)$ are used. Assuming the axiom of analytic determinacy,we prove that separable Banach spaces fall into four categories, namely:$\mathcal{K}(B_X)$ is equivalent either to a singleton, or to $\omega^\omega$,or to the family $\mathcal{K}(\mathbb{Q})$ of compact subsets of the rationalnumbers, or to the family $[\mathfrak{c}]^{<\omega}$ of all finite subsets ofthe continuum. Also under the axiom of analytic determinacy, a similarclassification of $\mathcal{AK}(B_X)$ is obtained. For separable Banach spacesnot containing $\ell^1$, we prove in ZFC that $\mathcal{K}(B_X) \sim\mathcal{AK}(B_X)$ are equivalent to either $\{0\}$, $\omega^\omega$,$\mathcal{K}(\mathbb{Q})$ or $[\mathfrak{c}]^{<\omega}$. The lattice structureof the family of all weakly null subsequences of an unconditional basis is alsostudied.

Tukey classification of some ideals in $\omega$ and the lattices of weakly compact sets in Banach spaces