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 concrete categories, i.e. categories equipped with faithful functor to Set

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

Then consider pairs of objects (aA,bB)(a \in A, b \in B) with the same underlying set, U(a)V(b)U(a) \simeq V(b). Then … (see the references Dimov-Tholen below).

Examples

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 June 12, 2014 16:06:37 by Colin Zwanziger (174.63.87.107)