# Amnestic functors

## Definition

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)