Typing tensor calculus in 2-categories (I)

To formalize calculations in linear algebra for the development of efficient algorithms and a framework suitable for functional programming languages and faster parallel computations, we adopt an approach that treats linear algebraic structures, such as matrices, as morphisms in the category of matrices, $\mathrm{Ma}{t}_k$. We further generalize this framework to arbitrary monoidal semiadditive categories. To extend this perspective and incorporate higher-rank matrices (tensors), we introduce the notion of semiadditive 2-categories, where matrices $T_{ij}$ are interpreted as 1-morphisms and tensors with four indices $T_{ijkl}$ as 2-morphisms. This formalization provides an index-free, typed framework for linear algebra that naturally accommodates matrices and tensors with up to four indices. Moreover, we extend the framework to monoidal semiadditive 2-categories and demonstrate explicit operations and vectorization techniques within the 2-category of 2Vec, as introduced by Kapranov and Voevodsky.