I know there are matrix rings, but what do we call the structure of the set of ALL real matrices, with the usual sum and product? We are looking for a kind of algebra whose operations aren't defined for each pair from the base set.
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7$\begingroup$ Linear category? $\endgroup$– Sam HopkinsCommented Jul 3, 2022 at 1:42
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7$\begingroup$ To spell out Sam Hopkins comment, a linear category is a category where each $\text{Hom}(A,B)$ is equipped with the structure of a $k$-vector space, and composition $\text{Hom}(A,B) \times \text{Hom}(B,C) \longrightarrow \text{Hom}(A,C)$ is bilinear. The category whose objects are the vector spaces $\mathbb{R}^1$, $\mathbb{R}^2$, $\mathbb{R}^3$, etcetera, with $\text{Hom}(\mathbb{R}^n, \mathbb{R}^m)$ being the $m \times n$ matrices is thus a linear category. $\endgroup$– David E SpeyerCommented Jul 3, 2022 at 1:55
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$\begingroup$ It's also a small category, since objects form a set. Hence a small linear category. $\endgroup$– YCorCommented Jul 3, 2022 at 7:35
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