Higher category theory, particularly the theory of infinite-dimensional categories (∞-categories), provides a powerful framework for modern mathematics and has found deep applications in theoretical physics. This paper challenges the hypothesis that ∞-categories can be physically realized. We argue that any physical system, being constrained by a finite information capacity as dictated by the holographic principle, cannot instantiate the infinite hierarchy of distinguishable morphisms characteristic of a true ∞-category. Our argument synthesizes foundational principles of quantum information theory and thermodynamics. We formalize this argument by proving that the information required to specify an infinite hierarchy of physically distinct morphisms would violate the Bekenstein bound on entropy. We present a theorem establishing that any physical system occupying a finite volume with finite energy must be describable by an n-category for some finite integer n. This conclusion holds even if the ∞-category admits a finite mathematical description, as the constraint applies to the physical instantiation of the states, not the complexity of the underlying rules. This suggests that while ∞-categories are an indispensable mathematical tool, they should be understood as formalisms for systems of very high, yet finite, complexity, imposing a fundamental limit on the organizational depth of the cosmos.
Pearl Bipin (Fri,) studied this question.