Categories of matrices
Posted on February 11, 2026 by Parth
(algebra)

A first class on category theory typically introduces this rather silly category whose objects are natural numbers, and morphisms \(p\to q\) are \(q\times p\) matrices – this is perhaps to drive home the point that anything and everything under the sun can be a category – and then proceeds to show that it is equivalent to the category of vector spaces (an “honest” category by anyone’s definition). Continuing this rather silly exercise, we will examine a category whose objects are all finite matrices, and then proceed to take off its guise.

Take a category \(\mathcal{C}\) whose objects are all (finite) matrices over a field \(\mathbb{k}\), with morphisms \(M\to N\) given by pairs of matrices \((F_1,F_2)\) such that \(MF_1=F_2N\). Note the shapes of \(F_1\) and \(F_2\) are determined by this relation, so the morphism-sets are \(\mathbb{k}\)-vector spaces in the obvious way.

Turns out the category \(\mathcal{C}\) is Abelian! A morphism \((F_1,F_2)\) is injective if and only if both \(F_1\) and \(F_2\) have no nullity; it is surjective if both have full rank.

We describe the category in two equivalent but more familiar forms – first, our category is evidently equivalent to \(\mathop{\mathsf{Rep}}(\bullet\to\bullet)\), i.e. the category of functors \([\bullet\to\bullet]\longrightarrow \mathbb{k}\text{-}\mathsf{mod}\). To spell this out, the objects of this category are diagrams of vector spaces \([V_1\xrightarrow{m} V_2]\), with morphisms \([V_1\xrightarrow{m}V_2]\longrightarrow [W_1\xrightarrow{n}V_2]\) given by commuting squares. There is a fully faithful functor \(\mathcal{C}\longrightarrow \mathop{\mathsf{Rep}}(\bullet\to \bullet)\) sending an \(p\times q\) matrix \(M\) to the corresponding linear map \(\mathbb{k}^q\to \mathbb{k}^p\), this functor is essentially surjective since every finite dimensional vector space has a basis.

Now given \([V_1\xrightarrow{m}V_2]\in\mathop{\mathsf{Rep}}(\bullet\to\bullet)\), consider the right \(A=\left(\begin{smallmatrix}\mathbb{k}&0\\\mathbb{k}&\mathbb{k}\end{smallmatrix}\right)\)-module with underlying vector space \(V_1\oplus V_2\), and \(A\)-action \[(v_1\oplus v_2)\cdot \left(\begin{smallmatrix}\lambda&0\\\xi&\mu\end{smallmatrix}\right)=(\lambda \cdot v_1+\xi\cdot mv_2 )\oplus(\mu \cdot v_2).\] Conversely any finitely generated \(A\)-module \(V\) has underlying vector space \(V\left(\begin{smallmatrix}1&0\\0&0\end{smallmatrix}\right) \oplus V\left(\begin{smallmatrix}0&0\\0&1\end{smallmatrix}\right)\), and the data of the linear map \[(-)\cdot\left(\begin{smallmatrix}0&0\\1&0\end{smallmatrix}\right)\colon V\left(\begin{smallmatrix}0&0\\0&1\end{smallmatrix}\right)\to V\left(\begin{smallmatrix}1&0\\0&0\end{smallmatrix}\right)\] determines \(V\) completely as an \(A\)-module. Thus in fact \(\mathcal{C}\simeq \mathop{\mathsf{Rep}}(\bullet\to\bullet)\) is equivalent to the \(\mathbb{k}\)-linear Abelian category of finitely generated right \(A\)-modules.

Generalising the above, the category of right modules over \(K_n=\left(\begin{smallmatrix}\mathbb{k}^{\phantom{n}}&0\\\mathbb{k}^n&\mathbb{k}\end{smallmatrix}\right)\) is equivalent to the category of representations of the \(n\)-Kronecker quiver, i.e. the category of \(n\)-tuples of equi-shape matrices \((M_1,...,M_n)\) where morphisms \((M_1,...,M_n)\to (N_1,...,N_n)\) are given by pairs of matrices \((F_1,F_2)\) such that \(M_iF_1=F_2N_i\) for all \(i\).