A weak representation of a functor$P : C^o \to Set$ (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 $\beta$ equation of a universal property and not the $\eta$ equation.

Definition

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

An object $X$ in $C$

natural transformations$s : P(\Gamma) \to C(\Gamma, X)$ and $r : C(\Gamma, X) \to P(\Gamma)$ such that $r$ is a retract of $s$.

Equivalently, by the Yoneda lemma, a weak representation is

An object $X$

A “weak universal morphism” $\epsilon : P(X)$

A natural transformation $I : P(\Gamma) \to C(\Gamma, X)$ that is a section of $P(-)(\epsilon) : C(\Gamma, X) \to P(\Gamma)$

Relation to Representable Functors

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

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