Affinizations and \(\boldsymbol{R}\)-matrices are one of the most powerful tools in the representation theory of quiver Hecke algebras and affine quantum groups. \(\boldsymbol{R}\)-matrices are distinguished homomorphisms between tensor products of modules, which measure the commutativity of tensor products. Affinizations of modules help \(\boldsymbol{R}\)-matrices to play their role. The study on affinizations and \(\boldsymbol{R}\)-matrices gives rise to the integer invariants \(\Lambda\), \(\widetilde{\Lambda}\)and \(\mathfrak{d}\), which have been used crucially in deriving several remarkable results including monoidal categorification of cluster algebras [S.-J. Kang et al., Compos. Math. 151, No. 2, 377–396 (2015; Zbl 1366.17014); J. Am. Math. Soc. 31, No. 2, 349–426 (2018; Zbl 1460.13039); M. Kashiwara et al., Adv. Math. 328, 959–1009 (2018; Zbl 1437.17005); M. Kashiwara and E. Park, J. Eur. Math. Soc. (JEMS) 20, No. 5, 1161–1193 (2018; Zbl 1475.16024)]. In the representation theory of quantum affine algebras, \(\boldsymbol{R}\)-matricesalready occupy a significant position. The generalized Schur-Weyl duality functor relates the \(\boldsymbol{R}\)-matricesin quiver Hecke algebras and the ones in quantum affine algebras in a natural way, and they enjoy very similar properties in their own categories [V. Chari and A. Pressley, A guide to quantum groups. Cambridge: Cambridge University Press (1994; Zbl 0839.17009); P. I. Etingof and A. A. Moura, Represent. Theory 7, 346–373 (2003; Zbl 1066.17005); M. Kashiwara, Duke Math. J. 112, No. 1, 117–195 (2002; Zbl 1033.17017); S.-J. Kang et al., Invent. Math. 211, No. 2, 591–685 (2018; Zbl 1407.81108); S.-J. Kang et al., Invent. Math. 216, No. 2, 597–599 (2019; Zbl 1447.81140); M. Kashiwara et al., Compos. Math. 156, No. 5, 1039–1077 (2020; Zbl 1497.17020), M. Kashiwara et al., Invent. Math. 236, No. 2, 837–924 (2024; Zbl 07828037)].
This paper establishes the theory of affinization and \(\boldsymbol{R}\)-matrices in the language of pro-objects, and as an application, constructs reflection functors over the localizations of quiver Hecke algebras of an arbitrary finite type. The main results of the paper goes as follows.

The first main result

is to study the notion of affinization and \(\boldsymbol{R}\)-matrices in the language of pro-objects. The authors define affinizations and \(\boldsymbol{R}\)-matrices in the categories with certain conditions, investigating their properties. Key properties appearing in quiver Hecke algebras and quantum affine algebras are rediscovered in a purely categorical setting. The authors next introduce a duality datum at the category level, constructing a generalized Schur-Weyl duality in the category setting after [M. Kashiwara et al., Invent. Math. 236, No. 2, 837–924 (2024; Zbl 07828037)].

The second main result

is to apply the generalized Schur-Weyl duality to the localized category \(\widetilde{\mathcal{C}^{\ast}}_{s_{i},w_{0}}\)of a quiver Hecke algebra of arbitrary finite type. As a result, for any \(i\in I\), the authors obtain an equivalence of monoidal categories \[ \mathcal{S}_{i}:\widetilde{\mathcal{C}}_{s_{i},w_{0}}\overset{\sim }{\rightarrow}\widetilde{\mathcal{C}^{\ast}}_{s_{i},w_{0}} \] which categorifies the braid group action [G. Lusztig, Introduction to quantum groups. Boston, MA: Birkhäuser (1993; Zbl 0788.17010), Chapter 37] and the Saito reflection \(\sigma_{i}\)[Y. Saito, Publ. Res. Inst. Math. Sci. 30, No. 2, 209–232 (1994; Zbl 0812.17013)].