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In this paper we apply presheaves to develop an invariant that can distinguish diffeomorphism classes of quasitoric manifolds in the category of quasitoric pairs Q. The objects in this category are pointed topological spaces (M, p) where M is a quasitoric manifold and p is a fixed point under the torus action. Maps between pairs are continuous, base-point preserving with respect to a certain topology that depends on the submanifolds of M . It is shown that the category of quasitoric manifoldsdoi:10.4310/hha.2019.v21.n1.a14 fatcat:impvktl6nrdidfnpxe6y4xgcyq