We study the category \(\mathcal {C}\) generated by extremal weight modules over \(U_q(\mathfrak {gl}_{>0})\) . We show that \(\mathcal {C}\) is a tensor category, and provide an explicit description of the socle filtration of tensor product of any two extremal weight modules. This follows from the study of Fock space \(\mathcal {F}^{\infty }\otimes \mathcal {M}\) of infinite level, which admits commuting actions of a parabolic q-boson algebra and \(U_p(\mathfrak {gl}_{>0})\) with \(p=-q^{-1}\) . Its socle has a duality, which can be viewed as a limit of level-rank duality on the fermionic Fock space \(\mathcal {F}^n\) of level n. To describe the socle filtration of \(\mathcal {F}^{\infty }\otimes \mathcal {M}\) , we introduce the notion of a saturated crystal valuation, whose existence was observed for example in the embedding of an extremal weight module into a tensor product of fundamental weight modules of affine type due to Kashiwara and Beck-Nakajima.
