nLab weak representation of a functor

Weak representation of a functor

Weak representation of a functor


A weak representation of a functor P:C oSetP : C^o \to Set (presheaf) is like a representable functor but only satisfying the representability condition up to retract rather than isomorphism. Unlike a representable, this structure is not unique up to isomorphism, and so we speak of representations as structure rather than representability as a property.

In type theoretic terms, this is an object that only satisfies the β\beta equation of a universal property and not the η\eta equation.


A weak representation of a presheaf P:C oSetP : C^o \to Set consists of

  1. An object XX in CC
  2. natural transformationss:P(Γ)C(Γ,X)s : P(\Gamma) \to C(\Gamma, X) and r:C(Γ,X)P(Γ)r : C(\Gamma, X) \to P(\Gamma) such that rr is a retract of ss.

Equivalently, by the Yoneda lemma, a weak representation is

  1. An object XX
  2. A “weak universal morphism” ϵ:P(X)\epsilon : P(X)
  3. A natural transformation I:P(Γ)C(Γ,X)I : P(\Gamma) \to C(\Gamma, X) that is a section of P()(ϵ):C(Γ,X)P(Γ)P(-)(\epsilon) : C(\Gamma, X) \to P(\Gamma)

Relation to Representable Functors

A weak representation (X,ϵ,I)(X, \epsilon, I) of PP induces an idempotent ee on the representable sets C(,X)C(-, X) such that PP is the idempotent splitting of the representable C(,X)C(-,X) in the presheaf category. By the Yoneda lemma, this induces an idempotent ee on XX in CC, and an object YY represents PP if and only if YY is the idempotent splitting of ee. As a consequence, a representable YY has a canonical section to any weak representation XX.

In type theoretic terms, the idempotent performs η \eta expansion, and so this says that a representable can be constructed from a weak representable as the fixed points of η\eta expansion, or dually as a quotient equating the η\eta expansion with the identity.

Created on August 19, 2022 at 14:05:20. See the history of this page for a list of all contributions to it.