Square root crystals and the square root of $B(\infty)$

We introduce a general monoidal category of $\mathbf{N}$-root crystals and then study the special case of square root $\mathfrak{gl}_n$-crystals. The latter objects include Yu's crystals on semistandard set-valued tableaux. Prior work of the first author, Tong, and Yu showed that regular square root $\mathfrak{gl}_n$-crystals can be a useful tool for proving Grothendieck positivity results. The objects studied here go beyond the regular case and allow us to construct a square root analog of the direct limit crystal $B(\infty)$. We give several descriptions of our square root of $B(\infty)$, using marginally large tableaux, the Lusztig or PBW parameterization, and the Nakashima--Zelevinsky polyhedral model. We show that this crystal has a simple character formula, exhibits a nontrivial Demazure filtration, and recovers Yu's semistandard set-valued tableau crystals after taking appropriate tensor products. We also investigate a number of differences between square root crystals and classical crystal constructions.