nLab Prof

Definition

$\mathbf{Prof}$ is the 2-category of categories, profunctors, and natural transformations.

Recall that a profunctor from $A$ to $B$ is a functor $B^{op}\times A\to Set$. Composition of profunctors in $Prof$ is by the “tensor product of functors” coend construction: if $H\colon A\to B$ and $K\colon B\to C$, their composite is given as a functor $C^{op}\times A \to Set$ by

$$(c,a)\mapsto \int^{b\in B} H(b,a)\times K(c,b).$$

The identity on a category $A$ is its hom-functor $Hom_A(-,-)$.

$\mathbf{Prof}$ can also be obtained as the Kleisli 2-category of PSh.

Properties

If profunctors are categorified binary relations, then $Prof$ is a categorification of Rel.

Note that as defined here, $Prof$ is a weak $2$-category or bicategory. A naturally defined equivalent strict 2-category has the same objects, but the morphisms $A\to B$ are cocontinuous functors $P A \to P B$, where $P A$ is the presheaf category of $A$. This is equivalent because a profunctor $A\to B$ can equivalently be regarded as a functor $A\to P B$, and $P A$ is the free cocompletion of $A$. This equivalence is an instance of Lack's coherence theorem.

Note that every functor $f\colon A\to B$ gives two representable profunctors $B(f-,-)$ and $B(-,f-)$. This defines two 2-functors $Cat \to Prof$ that are the identity on objects. The relationship between Cat and $Prof$ encoded in this way makes them into an equipment.

$Prof$ is a sort of classifying object for arbitrary functors; see displayed category.

There are also enriched and internal versions of $Prof$. These accordingly refine categories of enriched categories.

$Prof$ is a locally cocomplete bicategory.

Related concepts

• Pos

• Set

  • Rel, Span, Prof

• Grpd, ∞Grpd

• Cat,

  • Operad

  • VCat

• 2Cat

• (∞,1)Cat

  • (∞,1)Operad

• (∞,n)Cat

References

• Maren Justesen, Bikategorien af Profunktorer, Aarhus 1968 (pdf)

category: category