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Isbell envelope

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The Isbell Envelope of a Category

Idea

Given a category 𝒯\mathcal{T} of “test objects”, one can consider “things” which 𝒯\mathcal{T} can detect. If one already has a class of objects in mind, one can consider 𝒯\mathcal{T}-structures on those objects by looking at morphisms to and from objects of 𝒯\mathcal{T}. Then one considers questions only in so far as they can be distinguished after mapping into 𝒯\mathcal{T}. A simple example is that of smooth manifolds where many questions are solved by transporting the problem to Euclidean spaces via charts. Another example is the use of weak topologies on locally convex topological spaces. Without a class of objects in mind, the natural definition of an object probeable by 𝒯\mathcal{T}-objects involves presheaves and copresheaves and is as follows.

Definition

Let 𝒯\mathcal{T} be an essentially small category. Then the Isbell envelope of 𝒯\mathcal{T}, written E(𝒯)E(\mathcal{T}), is defined as follows. An object is a triple X=(P,F,c)X = (P,F,c) where

  1. PP is a contravariant functor 𝒯Set\mathcal{T} \to Set (a presheaf),
  2. FF is a covariant functor 𝒯Set\mathcal{T} \to Set (a copresheaf),
  3. c:P×F𝒯(,)c : P \times F \to \mathcal{T}(-,-) is a natural transformation of bifunctors 𝒯 op×𝒯\mathcal{T}^{op} \times \mathcal{T} \to Set.

(We conventionally write X=(P X,F X,c X)X = (P_X, F_X, c_X).) A morphism XYX \to Y is a pair of natural transformations (α,β)(\alpha,\beta) with α:P XP Y\alpha : P_X \to P_Y β:F YF X\beta : F_Y \to F_X which satisfy the relation c X(,β)=c Y(α,)c_X(-,\beta -) = c_Y(\alpha -,-).

The requirement that 𝒯\mathcal{T} be essentially small implies that the collections of natural transformations form sets, and thus that this is a locally small category.

Certain elementary properties are easy to prove.

Proposition

There is a (“double/twosided Yoneda”) embedding of 𝒯\mathcal{T} as a full subcategory of E(𝒯)E(\mathcal{T}) via

(1)T(𝒯(,T),𝒯(T,),) T \mapsto (\mathcal{T}(-,T), \mathcal{T}(T,-), \circ)

Identifying 𝒯\mathcal{T} with its image, there are natural isomorphisms

(2)𝒯 g(T,X)P X(T),𝒯 g(X,T)F X(T) \mathcal{T}^g(T,X) \cong P_X(T), \qquad \mathcal{T}^g(X,T) \cong F_X(T)

Other elementary properties to follow

Profunctors

The Isbell envelope of a category can be viewed as a category of profunctors. In short, the Isbell envelope of 𝒯\mathcal{T} consists of the lax factorisations of HomHom through 11.

Let us spell this out. Recall that a profunctor 𝒜\mathcal{A} \to \mathcal{B} is a functor 𝒜× opSet\mathcal{A} \times \mathcal{B}^{op} \to Set. Both covariant and contravariant functors to SetSet are special examples of profunctors: a covariant functor 𝔉:𝒜Set\mathfrak{F} : \mathcal{A} \to Set is a profunctor 𝒜1\mathcal{A} \to 1, where 11 is the terminal category, and a contravariant functor 𝔊:Set\mathfrak{G} : \mathcal{B} \to Set is a profunctor 11 \to \mathcal{B}. The composition of these as profunctors produces the obvious profunctor 𝔉×𝔊:𝒜× opSet\mathfrak{F} \times \mathfrak{G} : \mathcal{A} \times \mathcal{B}^{op} \to Set. Thus extracting the functor part of the definition of an object in the Isbell envelope of 𝒯\mathcal{T} produces a profunctor 𝒯𝒯\mathcal{T} \to \mathcal{T} which factors through 11.

There is an obvious profunctor 𝒯𝒯\mathcal{T} \to \mathcal{T} given by the HomHom-bifunctor. This is the identity for profunctor composition. The natural transformation from the definition of an object in the Isbell envelope of 𝒯\mathcal{T} defines a morphism of profunctors from the corresponding profunctor to HomHom. Thus an object in the Isbell envelope of 𝒯\mathcal{T} corresponds to an object in the subcategory of the slice category of profunctors over HomHom of those objects which factor through 11.

In other words, a lax factorisation of HomHom through 11.

Concrete Envelopes

A variant of the above involves a background category, say 𝒰\mathcal{U}. The test objects should be viewable also as objects of 𝒰\mathcal{U}, usually via a faithful functor. Any object of 𝒰\mathcal{U} defines a profunctor 𝒯𝒯\mathcal{T} \to \mathcal{T} (which factors through 11) via

(3)𝒯×𝒯 op 𝒰×𝒰 op Set (T 1,T 2) (T 1,T 2) 𝒰(U,T 1)×𝒰(T 2,U) \begin{aligned} \mathcal{T} \times \mathcal{T}^{op} &\to \mathcal{U} \times \mathcal{U}^{op} &&\to Set \\ (T_1, T_2) &\mapsto (|T_1|, |T_2|) &&\mapsto \mathcal{U}(U,|T_1|) \times \mathcal{U}(|T_2|,U) \end{aligned}

Then one can consider those objects of the Isbell envelope of 𝒯\mathcal{T} that are sub-profunctors of one of this type. In addition one should restrict the morphisms to those that are induced by morphism of objects of 𝒰\mathcal{U}. Translating this back to the language of functors yields the following definition.

Definition

Let 𝒯\mathcal{T} and 𝒰\mathcal{U} be categories with a faithful functor 𝒯𝒰\mathcal{T} \to \mathcal{U} which we shall write as TTT \mapsto |T|. The 𝒰\mathcal{U}-concrete Isbell envelope of 𝒯\mathcal{T}, which we shall write E 𝒰(𝒯)E_{\mathcal{U}}(\mathcal{T}), is the category whose objects are triples (U,P,F)(U,P,F) where

  1. UU is an object of 𝒰\mathcal{U},
  2. P:𝒯SetP : \mathcal{T} \to Set is a subfunctor of the (contravariant) functor T𝒰(T,U)T \mapsto \mathcal{U}(|T|,U),
  3. F:𝒯SetF : \mathcal{T} \to Set is a subfunctor of the (covariant) functor T𝒰(U,T)T \mapsto \mathcal{U}(U,|T|),

such that the image of the natural transformation

(4)P×F𝒰(,), P \times F \to \mathcal{U}(|-|,|-|),

which comes from composition in 𝒰\mathcal{U}, lies in the image of the natural transformation 𝒯(,)𝒰(,).\mathcal{T}(-,-) \to \mathcal{U}(|-|,|-|).

A morphism in E 𝒰(𝒯)E_{\mathcal{U}}(\mathcal{T}) is a 𝒰\mathcal{U}-morphism on the underlying 𝒰\mathcal{U}-objects that defines natural transformations P XP YP_X \to P_Y and F YF XF_Y \to F_X.

Having a background category ensures that the size issues with natural transformations do not occur and so we can drop the requirement that 𝒯\mathcal{T} be essentially small.

Enrichment

These notions can be considered in the setting of enriched category theory by replacing Set wherever it occurs (explicitly or implicitly) by the enriching category.

Isbell Duality

Within the Isbell envelope of 𝒯\mathcal{T} one can consider various subcategories where the objects satisfy extra conditions. An obvious condition is that the presheaf is actually a sheaf. Another useful condition is that of Isbell duality.

Definition

An object XX of E(𝒯)E(\mathcal{T}) is said to be PP-saturated if the obvious natural transformations

(5)P X(T)NatTrans(F X,F T) P_X(T) \to NatTrans(F_X,F_T)

are isomorphisms. It is said to be FF-saturated if the obvious natural transformations

(6)F X(T)NatTrans(P X,P T) F_X(T) \to NatTrans(P_X,P_T)

are isomorphisms. It is said to satisfy Isbell duality if it is both PP- and FF-saturated.

Within E(𝒯)E(\mathcal{T}) one can consider the full subcategories of PP-saturated objects, of FF-saturated objects, and those satisfying Isbell duality. Clearly, the last is the intersection of the first two. There are idempotent functors onto the first two categories given by replacing one of P XP_X or F XF_X by the natural transformations of the other. An interesting question is to ask whether or not the obvious iteration stabilises after a finite number of steps (which would result in an object satisfying Isbell duality).

If the test category has, or “morally has”, a representable functor to SetSet then there is a strong relationship between saturation and concreteness.

Definition

A constant separator in a category is an object, say SS, with the property that if fg:ABf \ne g : A \to B then there is a constant morphism c:SAc : S \to A such that fcgcf c \ne g c.

As the constant morphisms form a two-sided ideal, any object C 0C_0 in a category 𝒞\mathcal{C} defines a covariant functor C 0:𝒞Set|-|_{C_0} : \mathcal{C} \to Set which on objects sends CC to the set of constant morphisms from C 0C_0 to CC. A morphism C 0C 0C_0 \to C_0' defines a natural transformation (in the opposite direction) between the corresponding functors. From the properties of constant morphisms one can easily deduce that two different morphisms between the same objects induce the same natural transformations between the functors. Thus if there are morphisms between two objects in both directions the two functors are naturally isomorphic.

The property of being a constant separator is clearly equivalent to the condition that this constant functor be faithful. Moreover, it is easy to show that in a non-trivial category, any two constant separators have morphisms between them in both directions and so induce naturally isomorphic functors. Indeed, not just naturally isomorphic but naturally naturally isomorphic in that there is a canonical choice of natural isomorphism.

Now we transfer this to the Isbell envelope of 𝒯\mathcal{T}.

Definition

Let X=(P,F,c)X=(P,F,c) be an object of E(𝒯)E(\mathcal{T}). An element αP(T)\alpha \in P(T), for TT an object of 𝒯\mathcal{T}, is said to be constant if ϕα𝒯(T,T)\phi \circ \alpha \in \mathcal{T}(T,T') is constant for all TT' objects of 𝒯\mathcal{T} and ϕF(T)\phi \in F(T').

Let us write X T|X|_T for the set of constant elements in P(T)P(T).

Lemma

Let T 0T_0 be an object in 𝒯\mathcal{T}. The assignment XX T 0X \to |X|_{T_0} is functorial and extends the assignment TT T 0T \to |T|_{T_0}. A 𝒯\mathcal{T}–morphism T 0T 1T_0 \to T_1 defines a natural transformation of functors.

Proof

Let X i=(P i,F i,c i)X_i = (P_i,F_i,c_i), i=1,2i = 1,2, be two objects in E(𝒯)E(\mathcal{T}). Let (α,β)(\alpha,\beta) be a morphism from the first to the second. Then α\alpha is a natural transformation P 1P 2P_1 \to P_2 so in particular defines a morphism of sets P 1(T 0)P 2(T 0)P_1(T_0) \to P_2(T_0). Let γP 1(U 0)\gamma \in P_1(U_0) be a constant element. Let TT be an object in 𝒯\mathcal{T} and ϕF 2(T)\phi \in F_2(T). We need to show that ϕ(αγ)\phi \circ (\alpha \circ \gamma) is a constant morphism. By the definition of a morphism in E(𝒯)E(\mathcal{T}), ϕ(αγ)=(ϕβ)γ\phi \circ (\alpha \circ \gamma) = (\phi \circ \beta) \circ \gamma and this latter is a constant morphism since γP 1(T 0)\gamma \in P_1(T_0) is a constant element.

For the extension, we observe that for γP T(T 0)\gamma \in P_T(T_0) then if γ\gamma is a constant morphism, c T(ϕ,γ)=ϕγc_T(\phi,\gamma) = \phi \circ \gamma is constant for all ϕ\phi; whilst if c T(ϕ,γ)c_T(\phi,\gamma) is constant for all ϕ\phi then in particular γ=c T(1 T,γ)\gamma = c_T(1_T,\gamma) is constant. That the morphisms correspond is obvious.

A 𝒯\mathcal{T}–morphism ψ:T 0T 1\psi : T_0 \to T_1 defines a map P(T 1)P(T 0)P(T_1) \to P(T_0) as PP is a contravariant functor. Let γP(T 1)\gamma \in P(T_1) be a constant element. Then for TT' an object in 𝒯\mathcal{T} and ϕF(U)\phi \in F(U'), ϕ(γψ)=(ϕγ)ψ\phi \circ (\gamma \circ \psi) = (\phi \circ \gamma) \circ \psi and ϕγ\phi \circ \gamma is a constant morphism in 𝒯(T 1,T)\mathcal{T}(T_1,T') so ϕ(γψ)\phi \circ (\gamma \circ \psi) is a constant morphism in 𝒯(T 0,T)\mathcal{T}(T_0,T'). Hence γψ\gamma \circ \psi is a constant element in P(T 0)P(T_0).

The result that the natural transformations depend only on the existence of a morphism does not carry over to this extended setting.

Andrew: Useful to have an example here, of course.

Let us fix a 𝒯\mathcal{T}–object T 0T_0. We have two bifunctors 𝒯×E(𝒯)Set\mathcal{T} \times E(\mathcal{T}) \to Set given by

(T,X)P X(T) (T,X) \mapsto P_X(T)

and

(T,X)Set(T T 0,X T 0) (T,X) \mapsto Set(|T|_{T_0},|X|_{T_0})

Let us, to simplify notation, identify TT with its image in E(𝒯)E(\mathcal{T}). Then P X(T)P_X(T) is naturally isomorphic to E(𝒯)(T,X)E(\mathcal{T})(T,X) so the fact that XX T 0X \mapsto |X|_{T_0} is a functor defines a natural transformation from the first to the second via

P X(T)E(𝒯)(T,X)Set(T T 0,X T 0) P_X(T) \cong E(\mathcal{T})(T,X) \to Set(|T|_{T_0}, |X|_{T_0})

In a similar fashion, we obtain a natural transformation

F X(T)E(𝒯)(X,T)Set(X T 0,T T 0). F_X(T) \cong E(\mathcal{T})(X,T) \to Set(|X|_{T_0},|T|_{T_0}).

We therefore obtain a functorial choice of underlying concrete object (with the underlying category being SetSet). We can refine the notion of concreteness slightly.

Definition

Let 𝒯\mathcal{T} be an essentially small category with a concrete separator, say T 0T_0. Let X=(P X,F X,c X)X = (P_X,F_X,c_X) be an object of E(𝒯)E(\mathcal{T}). We say that XX is PP–concrete if the map of sets

P X(T)Set(T T 0,X T 0) P_X(T) \to Set(|T|_{T_0},|X|_{T_0})

is injective for all 𝒯\mathcal{T}–objects, TT.

Similarly, we say that XX is FF–concrete if the map of sets

F X(T)Set(X T 0,T T 0) F_X(T) \to Set(|X|_{T_0}, |T|_{T_0})

is injective for all 𝒯\mathcal{T}–objects, TT.

Clearly, if it is both PP-concrete and FF-concrete then it is concrete. Changing the concrete separator does not alter concreteness.

Theorem

Let 𝒯\mathcal{T} be an essentially small category admitting a constant separator. Then for an object XX of E(𝒯)E(\mathcal{T}), the following hold:

  1. If XX is PP-saturated then it is PP-concrete,
  2. If XX is FF-saturated then it is FF-concrete,
  3. If XX satisfies Isbell duality then it is concrete.
Proof

Clearly the first and second statements imply the third.

Let us consider the first.

Let XX be an object of E(𝒯)E(\mathcal{T}) such that P X(T)=NatTrans(F X,F T)P_X(T) = NatTrans(F_X,F_T) for all 𝒯\mathcal{T}–objects TT. Let SS be a concrete separator in 𝒯\mathcal{T}. We shall write |-| for S|-|_S. Let TT be an object of 𝒯\mathcal{T}. Let αβP X(T)\alpha \ne \beta \in P_X(T). As P X(T)=NatTrans(F X,F T)P_X(T) = NatTrans(F_X,F_T), their inequality means that they differ as natural transformations. Thus there is some TT' for which α\alpha and β\beta induce different morphisms F X(T)F T(T)F_X(T') \to F_T(T'). Thus there is some ϕF X(T)\phi \in F_X(T') such that α T(ϕ)β T(ϕ)𝒯(T,T)\alpha_{T'}(\phi) \ne \beta_{T'}(\phi) \in \mathcal{T}(T,T').

As SS is a constant separator, there is thus a constant morphism δ:ST\delta : S \to T such that α T(ϕ)δβ T(ϕ)δ\alpha_{T'}(\phi) \circ \delta \ne \beta_{T'}(\phi) \circ \delta. Using composition notation, we rewrite this as ϕαδϕβδ\phi \circ \alpha \circ \delta \ne \phi \circ \beta \circ \delta. As δ\delta is a constant morphism, αδ\alpha \circ \delta and βδ\beta \circ \delta are constant elements of P X(S)P_X(S). Since ϕαδϕβδ\phi \circ \alpha \circ \delta \ne \phi \circ \beta \circ \delta, they must be different constant elements. Thus the maps α,β:TX\alpha,\beta : |T| \to |X| are different and so XX is PP–concrete.

The second statement is very similar.

Let XX be an object of E(𝒯)E(\mathcal{T}) such that F X(U)=NatTrans(P X,P T)F_X(U) = NatTrans(P_X,P_T) for all 𝒯\mathcal{T}–objects TT. Let SS be a concrete separator in 𝒯\mathcal{T}. Let TT be an object of 𝒯\mathcal{T}. Let ϕψF X(T)\phi \ne \psi \in F_X(T). As F X(T)=NatTrans(P X,P T)F_X(T) = NatTrans(P_X,P_T), their inequality means that they differ as natural transformations. Thus there is some TT' for which ϕ\phi and ψ\psi induce different morphisms P X(T)P T(T)P_X(T') \to P_T(T'). Thus there is some αP X(T)\alpha \in P_X(T') such that ϕ T(α)ψ T(α)𝒯(T,T)\phi_{T'}(\alpha) \ne \psi_{T'}(\alpha) \in \mathcal{T}(T',T).

As SS is a constant separator, there is thus a constant morphism δ:ST\delta : S \to T' such that ϕ T(α)δψ T(α)δ\phi_{T'}(\alpha) \circ \delta \ne \psi_{T'}(\alpha) \circ \delta. Using the composition notation, we rewrite this as ϕαδψαδ\phi \circ \alpha \circ \delta \ne \psi \circ \alpha \circ \delta. As δ\delta is a constant morphism, αδP X(S)\alpha \circ \delta \in P_X(S) is a constant element. We therefore have an element of X|X| which distinguishes between the induced maps from ϕ\phi and ψ\psi. Hence XX is FF–concrete.

Revised on January 12, 2012 22:26:01 by Tim Campion? (173.76.90.116)