nLab
dualizing object

Idea

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

$$T:A^{\mathrm{op}}\stackrel{{\mathrm{Hom}}_A(-,a)}{\to }B$$T : A^{op} \stackrel{Hom_A(-,a)}{\to} B

$$S:B^{\mathrm{op}}\stackrel{{\mathrm{Hom}}_B(-,a)}{\to }A.$$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 $A$ and $B$ be two concrete categories, i.e. categories equipped with faithful functor to Set

$$U:A\to \mathrm{Set}$$U : A \to Set

$$V:B\to \mathrm{Set}.$$V : B \to Set \,.

Then consider pairs of objects $(a\in A,b\in B)$ with the same underlying set, $U(a)\simeq V(b)$. Then … (see reference below).

Examples

Examples appear at

• Chu construction

• rational homotopy theory

References

• 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)