Profunctor Double Category

Replacing truth values by sets turns the matrix double category into the profunctor double category. Matrices become profunctors, the order-relation 2-cells become actual maps of sets, and becomes coend. This note records the 2-cell shape and the decategorification dictionary that reads the two structures off each other.

The 2-cell

Objects are categories; horizontal 1-cells are profunctors (functors ); vertical 1-cells are functors . A 2-cell has components

natural in , with whiskering by functors that reindex source and target. This is the Set-valued lift of the inequality , where a map of sets replaces a comparison of scalars.

The decategorification dictionary

Remark on reading one structure off the other

The matrix double category is the entry-by-entry decategorification of the profunctor one:

• ;
• coend ;
• Boolean truth values ;
• monoid one-object category;
• ordered semiring additive / -enriched category.
  The last line is the slogan a categorified ring is an additive category, where sums of scalars become biproducts and the order becomes actual morphisms.

Where the ground varies

Remark on the relation to enrichment and coends

Read right to left, coend composition of profunctors is the categorified matrix product, which is why the same triangle of enrichment, Grothendieck and coend governs both. The coend is the once scalars are sets, and the additive / -enriched base is exactly where “adding” morphisms makes sense.

References

• F. Loregian, “(Co)end Calculus” (Cambridge, 2021)