nLab
amnestic functor

Amnestic functors

Definition

An amnestic functor is a functor U:π’Ÿβ†’π’žU : \mathcal{D} \to \mathcal{C} (between strict categories) that reflects identity morphisms: more precisely, for any isomorphism f:aβ†’bf\colon a \to b in π’Ÿ\mathcal{D}, if Ua=UbU a = U b and Uf=id UaU f = id_{U a}, then a=ba = b and f=id af = id_a.

If we follow the principle of equivalence and refuse to state equations between objects, then the closest thing to this that we can say is that if Uaβ‰…UbU a \cong U b (say via g:Uaβ†’Ubg\colon U a \to U b) and Uf=g∘id UaU f = g \circ id_{U a} (which is a trivial hypothesis, since gg can simply be UfU f), then aβ‰…ba \cong b (say via h:aβ†’bh\colon a \to b) and f=h∘id af = h \circ id_a (which is a trivial conclusion, since hh can simply be ff.) Thus, this is really a property of strict functors between strict categories. Or to put it another way, if U:π’Ÿβ†’π’žU\colon \mathcal{D} \to \mathcal{C} is any functor, then there exist equivalent categories π’Ÿβ€²β‰ƒπ’Ÿ\mathcal{D}' \simeq \mathcal{D} and π’žβ€²β‰ƒπ’ž\mathcal{C}' \simeq \mathcal{C} such that the composite π’Ÿβ€²β†’π’Ÿβ†’Fπ’žβ†’π’žβ€²\mathcal{D}' \to \mathcal{D} \overset{F}\to \mathcal{C} \to \mathcal{C}' is amnestic.

Properties

  • An amnestic full and faithful functor is automatically an isocofibration, i.e. injective on objects: if UDβ€²=UDU D' = U D, then there is some isomorphism f:Dβ€²β†’Df : D' \to D in π’Ÿ\mathcal{D} such that Uf=id UDU f = id_{U D}, but then we must have f=id UDβ€²=id UDf = id_{U D'} = id_{U D}, so Dβ€²=DD' = D.

  • An amnestic isofibration has the following lifting property: for any object DD in π’Ÿ\mathcal{D} and any isomorphism f:Cβ†’UDf : C \to U D in π’ž\mathcal{C}, there is a unique isomorphism f˜:CΛœβ†’D\tilde{f} : \tilde{C} \to D such that Uf˜=fU \tilde{f} = f. Indeed, if fΛœβ€²:CΛœβ€²β†’D\tilde{f}' : \tilde{C}' \to D were any other isomorphism such that UfΛœβ€²=fU \tilde{f}' = f, then U(f˜ βˆ’1∘fΛœβ€²)=id CU (\tilde{f}^{-1} \circ \tilde{f}') = id_C, so we must have f˜=fΛœβ€²\tilde{f} = \tilde{f}'.

  • If the composite U∘KU \circ K is an amnestic functor, then KK 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 April 3, 2017 14:20:41 by Toby Bartels (64.89.54.58)