An amnestic functor is a functor $U : \mathcal{D} \to \mathcal{C}$ (between strict categories) that reflects identity morphisms: more precisely, for any isomorphism$f\colon a \to b$ in $\mathcal{D}$, if $U a = U b$ and $U f = id_{U a}$, then $a = b$ and $f = id_a$.

Properties

An amnestic full and faithful functor is automatically an isocofibration, i.e. injective on objects: if $U D' = U D$, then there is some isomorphism$f : D' \to D$ in $\mathcal{D}$ such that $U f = id_{U D}$, but then we must have $f = id_{U D'} = id_{U D}$, so $D' = D$.

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

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

AdΓ‘mek, Herlich and Strecker, The Joy of Cats, Chapter I, Definition 3.27.

Revised on February 21, 2013 16:07:17
by Todd Trimble
(98.208.182.196)