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We construct Grassmannian categories of infinite rank, providing an infiniteanalogue of the Grassmannian cluster categories introduced by Jensen, King, andSu. Each Grassmannian category of infinite rank is given as the category ofgraded maximal Cohen-Macaulay modules over a certain hypersurface singularity.We show that generically free modules of rank $1$ in a Grassmannian category ofinfinite rank are in bijection with the Pl\"ucker coordinates in an appropriateGrassmannian cluster algebra of infinite rank. Moreover, this bijection isstructure preserving, as it relates rigidity in the category to compatibilityof Pl\"ucker coordinates. Along the way, we develop a combinatorial formula tocompute the dimension of the $\mathrm{Ext}^1$-spaces between any twogenerically free modules of rank $1$ in the Grassmannian category of infiniterank.

Categories for Grassmannian cluster algebras of infinite rank