nLab
nucleus of a profunctor

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Definition

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}.

(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.

The nucleus of FF is the center of this adjunction. In the case where F=HomF = Hom, the nucleus is called the Isbell or 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.)

References

Last revised on September 3, 2015 at 18:31:19. See the history of this page for a list of all contributions to it.