nLab
amnestic functor

Definition

An amnestic functor is a functor $U:𝒟\to 𝒞$ (between strict categories) that reflects identity morphisms: more precisely, for any isomorphism $f:a\to b$ in $𝒟$, if $Ua=Ub$ and $Uf={\mathrm{id}}_{Ua}$, then $a=b$ and $f={\mathrm{id}}_a$.

Properties

• An amnestic full and faithful functor is automatically an isocofibration, i.e. injective on objects: if $UD\prime =UD$, then there is some isomorphism $f:D\prime \to D$ in $𝒟$ such that $Uf={\mathrm{id}}_{UD}$, but then we must have $f={\mathrm{id}}_{UD\prime }={\mathrm{id}}_{UD}$, so $D\prime =D$.

• An amnestic isofibration has the following lifting property: for any object $D$ in $𝒟$ and any isomorphism $f:C\to UD$ in $𝒞$, there is a unique isomorphism $\tilde{f}:\tilde{C}\to D$ such that $U\tilde{f}=f$. Indeed, if $\tilde{f}\prime :\tilde{C}\prime \to D$ were any other isomorphism such that $U\tilde{f}\prime =f$, then $U({\tilde{f}}^{-1}\circ \tilde{f}\prime )={\mathrm{id}}_C$, so we must have $\tilde{f}=\tilde{f}\prime$.

• If the composite $U\circ K$ is an amnestic functor, then $K$ is also amnestic.

Examples

• Any strictly monadic functor is amnestic. Conversely, any monadic functor that is also an amnestic isofibration is necessarily strictly monadic.

References

• Adámek, Herlich and Strecker, The Joy of Cats, Chapter I, Definition 3.27.