nLab nucleus of a profunctor

Definition

If $C$ and $D$ are enriched categories over a cosmos $V$, that is, a complete and cocomplete symmetric closed monoidal category, then a profunctor $F$, from $C$ to $D$, that is, a $V$-functor $D^{op} \otimes C\to V$, induces two adjoint $V$-functors: $F_{\ast}: (V^C)^{op} \to V^{D^{op}}$ and $F^{\ast}: V^{D^{op}} \to (V^C)^{op}$. Abstractly, one $V$-functor is given by currying to get a $V$-functor $C \to V^{D^{op}}$ and then using the universal property of the free completion; and the other $V$-functor is given by currying to get a $V$-functor $D^{op} \to V^C$, taking opposites to get a $V$-functor $D \to (V^C)^{op}$, and then using the universal property of the free cocompletion. We then have an adjunction:

$$(V^C)^{op}(F^{\ast} p, q) \cong V^{D^{op}}(p, F_{\ast} q).$$

In the case of the $Hom$ profunctor, $Hom: A^{op} \otimes A \to V$, of a $V$-enriched category $A$, this adjunction is known as Isbell duality or Isbell conjugation. In analogy with this case, the general adjunction induced by a $V$-profunctor $F$ is called the Isbell adjunction induced by $F$.

The nucleus of $F$ is the center of this adjunction. In the case where $F = Hom$, the nucleus is called the reflexive completion.

Examples with $F = Hom$

• $V = Ab$, the category of abelian groups. Let $k$ be a field, viewed as a one-object $Ab$-category. Both $[k^{op},Ab]$ and $[k,Ab]$ are the category of $k$-vector spaces, and both adjoints are the dual vector space construction. The nucleus of the profunctor, or reflexive completion $R(k)$ of $k$, is the category of $k$-vector spaces $V$ for which the canonical map $V \to V^{\ast \ast}$ is an isomorphism — in other words, the finite-dimensional vector spaces.

Examples with other profunctors

• $V$ = truth values. Given two sets, $A$ and $B$, and a relation (truth-valued profunctor) between them, the adjunction is known as a Galois connection, which restricts to the nucleus, a Galois correspondence.
• $V = \overline{\mathbb{R}} = ([-\infty, \infty], \geq, +)$. Let a real vector space, $W$, be considered as a discrete $\overline{\mathbb{R}}$-category, and consider the $\overline{\mathbb{R}}$-profunctor corresponding to evaluation between an element of $W$ and an element of its dual. Then the nucleus is composed of $\overline{\mathbb{R}}$-valued functions on $W$, and the duality expresses the Legendre-Fenchel transform?. (See Simon Willerton’s post.)

References

• Simon Willerton, Tight spans, Isbell completions and semi-tropical modules, (tac)
• Simon Willerton, The Nucleus of a Profunctor: Some Categorified Linear Algebra, blog post
• Simon Willerton, Galois Correspondences and Enriched Adjunctions, blog post
• Shen, Lili, and Dexue Zhang. Categories enriched over a quantaloid: Isbell adjunctions and Kan adjunctions (2013), (TAC)