We have defined parallel composition of matrix categories, and matrix profunctors, and we can now define the three-dimensional structure of categories and matrix categories.

A metalogic is a “bifibrant triple category without interchange”; parallel composition is “2-weak”, like that of a tricategory.

<aside> ⚠️ Why no interchange? Parallel composition is not a double functor. It is neither lax nor colax with respect to sequential composition of matrix profunctors.

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$$ (i\otimes m)\diamond (j\otimes n) \nleftrightarrow (i\diamond j)\otimes (m\diamond n) $$

This is due to the combination of strict and weak colimits: weak-to-strict (lax, left-to-right above) is not total, while strict-to-weak (colax, right-to-left above) is not well-defined.

Sequential composition is given by coequalizer, while parallel composition is given by codescent object. The former equates elements, while the latter creates an isomorphism.

So, sequence-of-parallel $(i\otimes m)\diamond (j\otimes n)$ contains composites with associators which cannot be expressed as a parallel-of-sequence composite $(i\diamond j)\otimes (m\diamond n)$.

Hence there is no transformation $(i\otimes m)\diamond (j\otimes n) \Rightarrow (i\diamond j)\otimes (m\diamond n)$.

Yet in the other direction, there is a dual obstruction.

Elements of $(i\diamond j)\otimes (m\diamond n)$ are parallel-composable pairs connected along an identity in $\langle f\circ g\rangle$ — yet as stated before, these are the associative identities $(\mathrm{f_0,g_0}) = (\mathrm{f_1,g_1})$, which may be witnessed by multiple distinct zig-zags; and each gives distinct actions on the matrix categories $\mathcal{P}$ and $\mathcal{S}$.

So a transformation $(i\diamond j)\otimes (m\diamond n) \Rightarrow (i\otimes m)\diamond (j\otimes n)$ would have to be independent of the choice of zig-zag, but without a specific relation in $\langle\mathbb{Y}\rangle$ there is no way to compose $i$ with $m$ and $j$ with $n$.

Hence parallel composition is not colax, either.

Thus, parallel composition is neither lax nor colax for sequential composition; there is no interchange between the two operations. Recall also that that the weave construction $\langle -\rangle$ is not lax nor colax. So while $\mathbb{C}\mathrm{at}$ and $\mathbb{M}\mathrm{at} \mathbb{C}\mathrm{at}$ are double categories, parallel composition of $\mathbb{C}\mathrm{at} \leftarrow \mathbb{M}\mathrm{at} \mathbb{C}\mathrm{at} \to \mathbb{C}\mathrm{at}$ is a structure on spans of span categories.