# Easy examples of dual objects in category theory

Could any of you provide me with

1. a definition of a dual object, in the context of category theory, which could be understood by, e.g., sophomores in college, and

2. examples of dual objects which could be understood by the audience in 1?

• The dual of a vector space is the obvious example. – WillO May 9 '15 at 0:03
• "Dual object" by itself is ambiguous (there are lots of dualities in mathematics, and even lots of constructions that deserve to be called "dual object"). Better to say "monoidal dual." I agree that vector spaces are a good example. – Qiaochu Yuan May 9 '15 at 1:12

Duality in the context of category theory usually refers to "opposite" notions. The opposite of a category is the category obtained by formally reversing all the arrows. So the dual of a categorical definition or theorem is the same definition or theorem realized in the opposite category.

For example, the dual of a monomorphism is an epimorphism, since epis are monos in the opposite category. Interpreted in the category of sets, this says that injections are dual to surjections.

The dual of an equalizer is a coequalizer. Interpreted in the category of vector spaces, for example, this means that kernels are dual to cokernels.

You also have a more sophisticated duality between space and quantity. Many categories of spaces can be realized as opposites of cartain algebraic categories. At a more elementary level, we have the heuristic that rings or algebras can be thought of as rings or algebras of functions on some space.

• Thnaks for your answer. Could you be a little more specific on a very simple example of your choice? what about vector spaces? – Javier Arias May 9 '15 at 13:23
• What would you like to know more about regarding vector spaces? – ಠ_ಠ May 9 '15 at 20:18
• the specifics of their duals – Javier Arias May 9 '15 at 21:59
• The dual of a vector space in the sense of linear algebra is not actually an example of this sort of "opposite" duality I discussed. The operation that assigns to each vector space it's dual space is an example of an internal hom functor. – ಠ_ಠ May 10 '15 at 9:26
• some other people here suggested the dual of a vector space as a dual category.....is that false then? – Javier Arias May 10 '15 at 13:28

Perhaps a not-so-motivated-but-"easy" example would be to do the Int construction over the category of sets and partial injective functions. You could first introduce traced monoidal categories (now finite dimensional vector spaces would be a nice example) and then show how the Int construction works.