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The Fukaya-Seidel category is classically defined in terms of a Kähler manifold X equipped with a holomorphic function W, with the relevant data encoded in the exact one-form dW. However, a natural generalization suggests itself: to replace dW with an arbitrary closed (but not necessarily exact) holomorphic one-form α on X. This more general version arises in many constructions in low-dimensional topology. In this talk, I will explore how the framework of the algebra of the infrared provides a means to understand this extension and illuminates the structure of the resulting category.