Just as a matrix category is a bimodule of weave double categories, a matrix profunctor is a bimodule of weave vertical profunctors, which coheres with the associators and unitors of the source and target matrix categories.

Let $\mathbb{X,Y,A,B}$ be categories, $\mathcal{Q}(\mathbb{X,Y})$ and $\mathcal{R}(\mathbb{A,B})$ be matrix categories.

Let $f:\mathbb{X}\,|\,\mathbb{A}$ and $g: \mathbb{Y}\,|\,\mathbb{B}$ be profunctors, determining weave profunctors $f \Leftarrow \langle f\rangle\Rightarrow f$ and $g\Leftarrow \langle g\rangle \Rightarrow g$.

A matrix profunctor $i(f,g): \mathcal{Q}(\mathbb{X,Y})\,|\, \mathcal{R}(\mathbb{A,B})$ is a span profunctor which forms a bimodule from $\langle f\rangle$ to $\langle g\rangle$, that coheres with the associators and unitors of $\mathcal{Q}$ and $\mathcal{R}$.

Hence a matrix profunctor is a span profunctor

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with precompose action by $\langle f\rangle$ and postcompose action by $\langle g\rangle$

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https://q.uiver.app/#q=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&embed

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which are natural with respect to the associators

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4-c.png

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4-l.png

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and natural with respect to the unitors.

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4u-l.png

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4u-r.png

We summarize the concept of matrix profunctor, ordered by dimension.

dim concept structure syntax
2 matrix profunctor span profunctor $i(f,g):\mathcal{Q}(\mathbb{X,Y})\,
3 left action span transformation $\odot_f: \langle f\rangle
\ast i\Rightarrow i$
right action span transformation $\odot_g: i\ast \langle g\rangle\Rightarrow i$
4 c.assoc. coherence equality $\overline{\mathrm{x}}\odot (Q\odot \overline{\mathrm{y}})
\rightrightarrows (\overline{\mathrm{a}}\odot R)\odot \overline{\mathrm{b}}$
l.assoc. coherence equality $\mathrm{(\overline{x}_1\circ \overline{x}_2)}\odot Q\rightrightarrows \mathrm{\overline{a}_1\odot (\overline{a}_2}\odot R)$
r.assoc. coherence equality $Q\odot (\mathrm{\overline{y}_1\circ \overline{y}_2})\rightrightarrows (R\odot \mathrm{\overline{b}_1)\odot \overline{b}_2}$
l.unit coherence equality $\mathrm{\overline{id}.X}\odot Q\rightrightarrows \mathrm{\overline{id}.A}\odot R$
r.unit coherence equality $Q\odot \mathrm{\overline{id}.Y}\rightrightarrows R\odot \mathrm{\overline{id}.B}$

<aside> ⚠️ The elements of $i$ can be understood as inferences — however, the “collage” of a matrix profunctor is not quite a logic:

the processes of $f$ and $g$ do not act on $\mathcal{Q}$ and $\mathcal{R}$, i.e. they do not bend in the visual language.

</aside>

We only know that the actions of $\mathcal{Q}(\mathbb{X,Y})$ and $\mathcal{R}(\mathbb{A,B})$ are natural with respect to the elements of $i$, i.e. the bends can “slide through”: