Profunctors - Wiki - Evan Patterson

Given categories

\mathcal{C}

and

\mathcal{D}

, a profunctor from

\mathcal{C}

\mathcal{D}

, also known as a bimodule or distributor, is a functor

\Phi: \mathcal{C}^{\mathrm{op}} \times \mathcal{D} \to \mathbf{Set}

. There is a bicategory of categories, profunctors, and natural transformations, and also a a double category of categories, functors, profunctors, and natural transformations. Enriched profunctors are also important: a $\mathcal{V}$ -profunctor is a

\mathcal{V}

-functor

\mathcal{C}^\mathrm{op} \times \mathcal{D} \to \mathcal{V}

Profunctors generalize functors in a manner analogous to how relations generalize functions. Specifically, a profunctor between discrete categories taking values

\mathbf{0} = \{\}

and

\mathbf{1} = \{*\}

is a binary relation, just as a functor between discrete categories is a function. Profunctors can thus be considered a categorification of relations.

Literature

Introductions and surveys

Borceux, 1994: Handbook of Categorical Algebra, Vol. 1, Sec. 7.8: Distributors
Bénabou, 2000, ed. Streicher: Distributors at work (pdf)
Fong & Spivak, 2019: Seven Sketches in Compositionality, Sec. 4.2: Enriched profunctors
- Covers profunctors between categories enriched over a commutative quantale

Miscellaneous topics

Pécsi, 2012: On Morita contexts in bicategories (doi)
- Defines a bridge between categories $\mathcal{C}$ and $\mathcal{D}$ as a category $\mathcal{H}$ with objects $|\mathcal{H}| = |\mathcal{C}| \sqcup |\mathcal{D}|$ , containing $\mathcal{C}$ and $\mathcal{D}$ as full subcategories
- A directed bridge is then identified with the collage of a profunctor
- Characterizes equivalence and Morita equivalence of categories as the existence of certain kinds of bridges (Theorems 3.2 and 3.6)
- See also: Pécsi, 2012, PhD thesis: Bridges and profunctors (pdf)
  - The thesis itself is written in Hungarian but there is an English summary

Applications

Schultz, Spivak, Vasilakopoulou, Wisnesky, 2017: Algebraic databases (pdf, arxiv)
- Formalizes relational databases as a double category of schemas, schema mappings, and schema bimodules, where both schemas and schema bimodules are profunctors with extra structure
- Sec. 2: Background material on profunctors and proarrow equipments