nLab matrix theory

for matrix theory in the sense of string theory see at BFSS matrix model and IKKT matrix model

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Let $A$ be a Lawvere theory with generic object $T$. For $n \in \mathbb{N}$, the full subcategory of $A$ generated by the cartesian powers of $T^n$ is also a Lawvere theory, which we denote by $M_n(A)$. In the case of an annular theory (the theory of modules over a ring that we also call $A$), this is the construction of $n\times n$ matrices over $A$. If we denote by $M_n$ the application of this construction to the initial theory (the theory of sets), then we may identify $M_n(A)$ with the tensor product theory $M_n\otimes A$.

It is an amusing exercise to present $M_n$ in terms of generating operations and relations between them.