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arXiv · 2601.05972

Categorical Foundations for CuTe Layouts

Jack Carlisle·Jay Shah·Reuben Stern·Paul VanKoughnett
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Abstract

NVIDIA's CUTLASS library provides a robust and expressive set of methods for describing and manipulating multi-dimensional tensor data on the GPU. These methods are conceptually grounded in the abstract notion of a CuTe layout and a rich algebra of such layouts, including operations such as composition, logical product, and logical division. In this paper, we present a categorical framework for understanding this layout algebra by focusing on a naturally occurring class of tractable layouts. To this end, we define two categories Tuple and Nest whose morphisms give rise to layouts. We define a suite of operations on morphisms in these categories and prove their compatibility with the corresponding layout operations. Moreover, we give a complete characterization of the layouts which arise from our construction. Finally, we provide a Python implementation of our categorical constructions, along with tests that demonstrate alignment with CUTLASS behavior. This implementation can be found at our git repository https://github.com/ColfaxResearch/layout-categories.

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Jack Carlisle, Jay Shah, Reuben Stern, Paul VanKoughnett. 2026-01-09. Categorical Foundations for CuTe Layouts. https://arxiv.org/abs/2601.05972

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