nLab
dualizing object

This entry is about a concept of duality in general category theory. For the concept of dualizing objects in a closed category as used in homological algebra and stable homotopy theory see at dualizing object in a closed category.

Contents

Idea

A dualizing object is an object aa which can be regarded as being an object of two different categories AA and BB, such that the concrete duality which is induced by homming into that object induces dual adjunctions between AA and BB, schematically:

T:A opHom A(,a)B T : A^{op} \stackrel{Hom_A(-,a)}{\to} B
S:B opHom B(,a)A. S: B^{op} \stackrel{Hom_B(-,a)}{\to} A \,.

Many famous dualities are induced this way, for instance Stone duality and Gelfand-Naimark duality.

Remark on terminology

There are various different terms for “dualizing objects”. As recalled on p. 112 of the article by Porst and Tholen below

  • Isbell speaks of objects keeping summer and winter homes;

  • Lawvere speaks of objects sitting in two categories;

  • Simmons speaks of schizophrenic objects.

It has been convincingly argued by Tom Leinster (blog comment here) that the term “schizophrenic” should not be used. Todd Trimble then suggested the term “ambimorphic object.” Another suggestion was “Janusian object.”

Definition

Let AA and BB be two categories over SetSet, given by functors

U:ASet U : A \to Set
V:BSet. V : B \to Set \,.

(Traditionally in what follows, one assumed UU and VV to be faithful, i.e., one considered AA and BB to be categories of sets with extra structure, aka concrete categories, although we won’t actually need this. Or, sometimes one sees the hypothesis that U,VU, V are representable, but we won’t require this either.)

The situation described in the Idea section asks us to consider contravariantly adjoint functors T:ABT: A \to B, S:BAS: B \to A such that US:BSetU S: B \to Set and VT:ASetV T: A \to Set are representable, i.e., are given by isomorphisms B(,b)USB(-, \mathbf{b}) \cong U S (with representing element ϕUSb\phi \in U S \mathbf{b}) and A(,a)VTA(-, \mathbf{a}) \cong V T (with representing element ψVTa\psi \in V T \mathbf{a}). Let η:1 AST\eta: 1_A \to S T and ϵ:1 BTS\epsilon: 1_B \to T S denote the universal maps of the adjunction (so that if we write TT and SS in covariant form as T:AB opT: A \to B^{op} and S:B opAS: B^{op} \to A, we would have TST \dashv S with η\eta as unit and ϵ\epsilon as counit).

Proposition

The representing data ϕUSb,ψVTa\phi \in U S \mathbf{b}, \psi \in V T \mathbf{a} induce a canonical isomorphism ω:UaVb\omega: U \mathbf{a} \stackrel{\sim}{\to} V \mathbf{b}. It is given by the evident composite

UaUηaUSTaB(Ta,b)Set(VTa,Vb)eval ψVb.U \mathbf{a} \stackrel{U\eta \mathbf{a}}{\to} U S T \mathbf{a} \stackrel{\sim}{\to} B(T \mathbf{a}, \mathbf{b}) \to Set(V T \mathbf{a}, V \mathbf{b}) \stackrel{eval_\psi}{\to} V \mathbf{b}.

The inverse is given by a similar composite but switching the roles of a\mathbf{a} and b\mathbf{b}, ϕ\phi and ψ\psi, η\eta and ϵ\epsilon, SS and TT, and UU and VV.

Proof

Letting xx be an element of UaU \mathbf{a}, the element ω(x)Vb\omega(x) \in V \mathbf{b} is by definition (Vg)(ψ)(V g)(\psi) where g:Tabg: T \mathbf{a} \to \mathbf{b} is the unique map (by representability) such that (Uηa)(x)=(USg)(ϕ)(U \eta \mathbf{a})(x) = (U S g)(\phi). The alleged inverse ω¯:VbUa\widebar{\omega}: V \mathbf{b} \to U \mathbf{a} takes an element yVby \in V \mathbf{b} to (Uf)(ϕ)(U f)(\phi), where f:Sbaf: S \mathbf{b} \to \mathbf{a} is the unique map such that (VTf)(ψ)=(Vϵb)(y)(V T f)(\psi) = (V \epsilon \mathbf{b})(y).

To check these maps are inverse, one may check simply (ωω¯)(y)=y(\omega \widebar{\omega})(y) = y for any yVby \in V \mathbf{b}; the other equation ω¯ω=id\widebar{\omega} \omega = id follows by symmetry. With notation as above, put x=ω¯(y)x = \widebar{\omega}(y), so x=(Uf)(ϕ)Uax = (U f)(\phi) \in U \mathbf{a}. Now we embark on a diagram chase, starting with

USb Uf Ua Uηa USTa ϕ UηSb USTf 1 USTSb B(Ta,b) ξ B(Tf,b) B(TSb,b) \array{ U S \mathbf{b} & \stackrel{U f}{\to} & U \mathbf{a} & \stackrel{U \eta \mathbf{a}}{\to} & U S T \mathbf{a} \\ _\mathllap{\phi} \uparrow & \searrow_\mathrlap{U \eta S \mathbf{b}} & & _\mathllap{U S T f} \nearrow & \uparrow_\mathrlap{\cong} \\ 1 & & U S T S \mathbf{b} & & B(T \mathbf{a}, \mathbf{b}) \\ & _\mathllap{\xi} \searrow & \uparrow_\mathrlap{\cong} & \nearrow _\mathrlap{B(T f, \mathbf{b})} \\ & & B(T S \mathbf{b}, \mathbf{b}) & & }

where the top and right quadrilaterals commute by naturality, and ξ:TSbb\xi: T S \mathbf{b} \to \mathbf{b} is the unique arrow making the left quadrilateral commute. In fact ξ\xi is the structure map for a TST S-algebra structure on b\mathbf{b}, although for our purposes we will only need the unit axiom for such a structure. Following the perimeter of this diagram, we see

g=(TaTfTSbξb)g = (T \mathbf{a} \stackrel{T f}{\to} T S \mathbf{b} \stackrel{\xi}{\to} \mathbf{b})

is the unique arrow g:Tabg: T \mathbf{a} \to \mathbf{b} such that (USg)(ϕ)=(Uηa)(x)(U S g)(\phi) = (U \eta \mathbf{a})(x). Continuing on our way, applying VV to gg and evaluating at ψ\psi, we have only to check that the composite

1ψVTaVTfVTSbVξVb1 \stackrel{\psi}{\to} V T \mathbf{a} \stackrel{V T f}{\to} V T S \mathbf{b} \stackrel{V \xi}{\to} V \mathbf{b}

equals y:1Vby: 1 \to V \mathbf{b}. But already we said (VTf)(ψ)=(Vϵb)(y)(V T f)(\psi) = (V \epsilon \mathbf{b})(y), so the last composite is

1yVbVϵbVTSbVξVb1 \stackrel{y}{\to} V \mathbf{b} \stackrel{V \epsilon \mathbf{b}}{\to} V T S \mathbf{b} \stackrel{V \xi}{\to} V \mathbf{b}

and all we need to do now is check the unit equation ξ(ϵb)=1 b\xi \circ (\epsilon \mathbf{b}) = 1_\mathbf{b}. But this follows from representability of USU S, applied to the diagram

USb UηSb USTSb USϵb USb ϕ 1 ξ B(TSb,b) B(ϵb,b) B(b,b)\array{ U S \mathbf{b} & \stackrel{U \eta S \mathbf{b}}{\to} & U S T S \mathbf{b} & \stackrel{U S \epsilon \mathbf{b}}{\to} & U S \mathbf{b} \\ _\mathllap{\phi} \uparrow & & \uparrow_\mathrlap{\cong} & & \uparrow_\mathrlap{\cong} \\ 1 & \underset{\xi}{\to} & B(T S \mathbf{b}, \mathbf{b}) & \underset{B(\epsilon \mathbf{b}, \mathbf{b})}{\to} & B(\mathbf{b}, \mathbf{b}) }

where the top horizontal composite is an identity, according to a triangular equation.

The canonical identification of sets ω:Ua=Vb\omega: U \mathbf{a} = V \mathbf{b} means that we have an object “sitting in two categories” (Lawvere), namely an AA-structure a\mathbf{a} on this set and a BB-structure b\mathbf{b} on the same set, with T:A opBT: A^{op} \to B providing a lift of A(,a):A opSetA(-, \mathbf{a}): A^{op} \to Set through V:BSetV: B \to Set, and S:B opAS: B^{op} \to A lifting B(,b):B opSetB(-, \mathbf{b}): B^{op} \to Set through U:ASetU: A \to Set.

Given functors U:ASetU: A \to Set, V:BSetV: B \to Set, we define Adj rep(U,V)Adj_{rep}(U, V) to be the category whose objects are contravariantly adjoint pairs (S:BA,T:AB)(S: B \to A, T: A \to B) such that USU S and VTV T are representable. Morphisms (S,T)(S,T)(S, T) \to (S', T') are pairs of natural transformations α:SS\alpha: S \to S', β:TT\beta: T \to T' such that the diagrams

1 A η ST TS Tα TS η Sβ βS ϵ ST αT ST TS ϵ 1 B\array{ 1_A & \stackrel{\eta'}{\to} & S' T' & & & & & & T S' & \stackrel{T \alpha}{\to} & T S \\ _\mathllap{\eta} \downarrow & & \downarrow_\mathrlap{S' \beta} & & & & & & _\mathllap{\beta S'} \downarrow & & \downarrow_\mathrlap{\epsilon} \\ S T & \underset{\alpha T}{\to} & S' T & & & & & & T' S' & \underset{\epsilon'}{\to} & 1_B }

commute. We define 2-Pull(U,V)2\text{-}Pull(U, V) to be the 2-pullback of UU and VV in CatCat (i.e., the bi-iso-comma-object). In conjunction with the Yoneda lemma, the preceding proposition can be read as giving the construction of a functor Φ:Adj rep(U,V)2-Pull(U,V)\Phi: Adj_{rep}(U, V) \to 2\text{-}Pull(U, V).

Here then is one key definition.

Definition

Given U:ASetU: A \to Set, V:BSetV: B \to Set, a (U,V)(U, V)-dualizing object (or (U,V)(U, V)-ambimorphic object) is a triple (a,b,ω:UaVb)(\mathbf{a}, \mathbf{b}, \omega: U \mathbf{a} \stackrel{\sim}{\to} V \mathbf{b}) in the essential image of Φ\Phi.

Naturally represented adjunctions

Definition 1 is reasonably general and is one of several notions of “schizophrenic object” given in Dimov-Tholen93. A somewhat tighter notion, also considered in Dimov-Tholen89 and Porst-Tholen and which covers many cases that arise in practice, involves initial lifts through the functors UU, VV.

Again, suppose given contravariant adjoint functors S,TS, T with USU S and VTV T representable as above. We have maps

γ(UUηUSTB(T,b)),δ(VVϵVTSA(U,a))\gamma \coloneqq (U \stackrel{U \eta}{\to} U S T \cong B(T-, \mathbf{b})), \qquad \delta \coloneqq (V \stackrel{V \epsilon}{\to} V T S \cong A(U-, \mathbf{a}))

so that for each object aa of AA and xUax \in U a, we have a corresponding map (γa)(x):Tab(\gamma a)(x): T a \to \mathbf{b} (playing the role of “gg” in the proof of Proposition 1), and similarly for each object bb of BB and yVby \in V b, we have a map (δb)(y):Sba(\delta b)(y): S b \to \mathbf{a} (playing the role of “ff” in Proposition 1).

Definition

The adjunction between SS and TT is naturally represented if the family {(γa)(x):Tab} xUa\{(\gamma a)(x): T a \to \mathbf{b}\}_{x \in U a} is VV-initial for each aOb(A)a \in Ob(A), and the family {(δb)(y):Sba} yVb\{(\delta b)(y): S b \to \mathbf{a}\}_{y \in V b} is UU-initial for each bOb(B)b \in Ob(B).

Proposition

The restriction of Φ:Adj rep(U,V)2-Pull(U,V)\Phi: Adj_{rep}(U, V) \to 2\text{-}Pull(U, V) to the full subcategory whose objects are naturally represented adjoint pairs is full and faithful.

See Dimov-Tholen, Proposition 2.3, where the proof is sketched. Thus naturally represented adjoint pairs could be equivalently described as certain types of ambimorphic objects. We now describe these.

Let us denote the action of AA on the module U:ASetU: A \to Set, with components A(a,a)×UaUaA(a, a') \times U a \to U a', by the notation (f,x)f Ux(f, x) \mapsto f \cdot_U x. Thus each xUax \in U a induces a map Ux:A(a,a)Ua- \cdot_U x: A(a, \mathbf{a}) \to U \mathbf{a}. Similarly, each yVby \in V b induces a map Vy:B(b,b)Vb- \cdot_V y: B(b, \mathbf{b}) \to V \mathbf{b}.

Definition

A (U,V)(U, V)-ambimorphic object (a,b,ω:UaVb)(\mathbf{a}, \mathbf{b}, \omega: U \mathbf{a} \stackrel{\sim}{\to} V \mathbf{b}) is natural if for every aOb(A)a \in Ob(A), the VV-structured source diagram

{A(a,a) UxUaωVb} xUa\{A(a, \mathbf{a}) \stackrel{- \cdot_U x}{\to} U \mathbf{a} \stackrel{\omega}{\to} V \mathbf{b}\}_{x \in U a}

admits an initial lift targeted at b\mathbf{b} (so a suitable diagram of the form γ x:Tab\gamma_x: T a \to \mathbf{b} indexed over xUax \in U a, for some object TaT a of BB), and for every bOb(B)b \in Ob(B), the UU-structured source diagram

{B(b,b) VyVbω 1Ua} yVb\{B(b, \mathbf{b}) \stackrel{- \cdot_V y}{\to} V \mathbf{b} \stackrel{\omega^{-1}}{\to} U \mathbf{a}\}_{y \in V b}

admits an initial lift (a diagram of type δ y:Sba\delta_y: S b \to \mathbf{a}), targeted at a\mathbf{a}.

Thus the category of naturally represented adjoint pairs relative to (U,V)(U, V) is equivalent to the category of natural (U,V)(U, V)-dualizing objects. This is the notion of “schizophrenic object” given by Porst-Tholen; a reasonably detailed proof of the equivalence with naturally represented adjoint pairs is given in their Theorem 1.7.

Examples

One easy general example of the notion given in Definition 1 is where CC is a symmetric monoidal closed category and dd is an object therein. Here we may take U=V=C(I,):CSetU = V = C(I, -): C \to Set where II is the monoidal unit, and the contravariant representable functor C(,d):CSetC(-, d): C \to Set lifts to a contravariant enriched hom [,d]:CC[-, d]: C \to C; the symmetry isomorphism can be exploited to show how [,d][-, d] is adjoint to itself. See here for more.

Further examples appear at

References

  • G. D. Dimov, W. Tholen, A Characterization of Representable Dualities, In: Categorical Topology and its Relation to Analysis, Algebra and Combinatorics, Prague, Czechoslovakia 22-27 August 1988, J. Adamek and S. MacLane (eds.), World Scientific, Singapore, New Jersey, London, Hong Kong, 1989, pp. 336-357.
  • G. D. Dimov, W. Tholen, Groups of Dualities, Trans. Amer. Math. Soc., 336 (2), 901-913, 1993. (pdf)
  • H.-E. Porst, W. Tholen, Concrete Dualities in Category Theory at Work, Herrlich, Porst (eds.) (pdf)
  • Michael Barr, John F. Kennison, R. Raphael, Isbell Duality (pdf)

Revised on January 25, 2015 03:07:13 by Todd Trimble (67.81.95.215)