nLab nucleus of a profunctor




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

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

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

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

Examples with F=HomF = Hom

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

Examples with other profunctors

  • VV = truth values. Given two sets, AA and BB, 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=¯=([,],,+)V = \overline{\mathbb{R}} = ([-\infty, \infty], \geq, +). Let a real vector space, WW, be considered as a discrete ¯\overline{\mathbb{R}}-category, and consider the ¯\overline{\mathbb{R}}-profunctor corresponding to evaluation between an element of WW and an element of its dual. Then the nucleus is composed of ¯\overline{\mathbb{R}}-valued functions on WW, and the duality expresses the Legendre-Fenchel transform?. (See Simon Willerton’s post.)


  • 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)

Last revised on October 26, 2023 at 09:38:53. See the history of this page for a list of all contributions to it.