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This is the first part of a series of two papers aiming to construct acategorification of the braiding on tensor products of Verma modules, and inparticular of the Lawrence--Krammer--Bigelow representations. \\ In this part,we categorify all tensor products of Verma modules and integrable modules forquantum $\mathfrak{sl_2}$. The categorification is given by derived categoriesof dg versions of KLRW algebras which generalize both the tensor productalgebras of Webster, and the dg-algebras used by Lacabanne, the second authorand Vaz. We compute a basis for these dgKLRW algebras by using rewritingmethods modulo braid-like isotopy, which we develop in an Appendix.

Categorification of infinite-dimensional $\mathfrak{sl}_2$-modules and braid group 2-actions I: tensor products