nLab
amnestic functor

Amnestic functors

Amnestic functors

Idea

A functor is amnestic if its domain has no more duplication of isomorphic objects than its codomain. This formalizes the sense in which (for example) the concrete category Met contMet_cont of metric spaces and continuous maps is often better thought of as the category MetTopMet Top of metrizable topological spaces (and continuous maps), by identifying a property that the forgetful functor MetTopSetMet Top \to Set has but Met contSetMet_cont \to Set does not. As ‘amnestic’ is basically a fancy synonym of ‘forgetful’, the idea is to identify a property that one would like in a forgetful functor.

There is a corresponding notion of amnesticization of a functor (called the amnestic modification in The Joy of Cats), which replaces the domain with an equivalent category, relative to which the functor becomes amnestic. Applying this to Met contSetMet_cont \to Set produces a category isomorphic to MetTopMet Top. Although not needed in this case, we need the axiom of choice (AC) in general to prove that every functor has an amnesticization. Without AC, we can use amnestic anafunctors to make everything work out, although much of the convenience is lost.

Amnesticity is really a property of strict functors (or anafunctors) between strict groupoids. Groupoids, because the non-isic morphisms play no role in the definition; only the categories' cores matter, and a functor is amnestic iff its core is amnestic. Strict, because the definition requires us to state (in two places) that some isomorphic objects are equal; weakening the definition to follow the principle of equivalence leads to a trivial property that every functor satisfies. (That is, up to equivalence, every functor is amnestic, which is because every functor is equivalent to its amnesticization.)

Definitions

Definition

Let CC and DD be two strict categories, and let UU be a strict functor from DD to CC. We say that UU is amnestic if its core reflects identity morphisms. Explicitly, UU is amnestic iff, whenever aa and bb are objects of DD, ff is an isomorphism in DD from aa to bb, U(a)U(a) and U(b)U(b) are equal objects in CC, and U(f)U(f) is the identity morphism on this object in CC, then aa and bb are equal objects in DD, and ff is the identity morphism on this object in DD.

If we follow the principle of equivalence and refuse to state equations between objects, then we must modify the hypothesis to say that U(a)U(a) and U(b)U(b) are isomorphic in CC (say via g:U(a)Ubg\colon U(a) \to U b) and U(f)U(f) is the identity relative to this isomorphism (so U(f)=id U(b)gid U(a)U(f) = \id_{U(b)} \circ g \circ \id_{U(a)}; since we can simply let gg be U(f)U(f), this is trivial (beyond the initial isomorphism f:abf\colon a \to b). Similarly, we must modify the conclusion to say that aa and bb are isomorphic (say via h:abh\colon a \to b) and ff is the identity relative to this isomorphism (so f=id bhid af = \id_b \circ h \circ \id_a); since we can simply let hh be ff, this is also trivial. Thus up to equivalence, this property is trivial; on the other hand, it is preserved by isomorphism.

Definition

Now let CC and DD be two strict categories, and let UU be an anafunctor from DD to CC. We again say that UU is amnestic if its core reflects identity morphisms. Explicitly, now UU is amnestic iff, whenever aa and bb are objects of DD, ff is an isomorphism in DD from aa to bb, α\alpha is a specification of aa for the anafunctor UU, β\beta is a specification of bb for the anafunctor UU, U α(a)U_\alpha(a) and U β(b)U_\beta(b) are equal objects in CC, and U α,β(f)U_{\alpha,\beta}(f) is the identity morphism on this object in CC, then aa and bb are equal objects in DD, and ff is the identity morphism on this object in DD.

We might also demand that α=β\alpha = \beta; this is automatic if UU is saturated.

Properties

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

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

  • If the composite UKU \circ K is an amnestic functor, then KK is also amnestic.

Examples

  • Most famous forgetful functors are amnestic, such as GrpSetGrp \to Set, TopSetTop \to Set, TopGrpGrpTop Grp \to Grp, and TopGrpTopTop Grp \to Top. Even MetSetMet \to Set is amnestic, since the morphisms in Met are short maps. However, Met contSetMet_cont \to Set, where the morphisms are continuous maps, is not amnestic, as is shown by any set with two different but topologically equivalent metrics (such as 2\mathbb{R}^2 with the l 1l^1 and l l^\infty metrics).

  • The forgetful functor from a groupoid of structured sets is amnestic. The examples above may all be defined by starting from such a groupoid and specifying which functions preserve the structure (and so are morphisms in the category of structured sets). So long as this produces no additional isomorphisms, the forgetful functor will be amnestic. But in the case of Met contMet_cont, any non-isometric homeomorphism will be an isomorphism that was not in the original groupoid (consisting only of surjective isometries), and so this forgetful functor is not amnestic.

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

References

  • J. Adámek, H. Herrlich and G.E. Strecker: The Joy of Cats, Chapter I, Definition 3.27.

  • G. Preuß: Theory of Topological Structures: An Approach to Categorical Topology. D. Reidel Publishing Company. Mathematics and Its Applications, Dordrecht, Holland 1988. p. 178, footnote 31

Last revised on August 18, 2019 at 21:32:24. See the history of this page for a list of all contributions to it.