ABSTRACT
A duality is an equivalence between a category A and the dual of a category B. As it is the case for every equivalence, a duality is useful since it allows to transport properties from one side to the other; the presence of the dual category on one side is useful since our knowledge of properties of a category is typically asymmetric. Arguably the most “famous” duality result is Stone’s duality theorem Stone ≃ Booleop stating that the category Stone of (nowadays called) Stone spaces and continuous maps is dually equivalent to the category Boole of Boolean algebras and homomorphisms. Stone’s theorem can be seen as the “mother” of many similar results [2]; in particular, motivated by questions in semantics of modal (propositional) logics, there is substantial ongoing research about extensions of the classical Stone duality to categories of co-algebras (for instance, [3] and [1]). In this talk we shall explain the general construction of dualities involving dualising objects [4], and show how the Kleisli construction for monads can simplify the proof and presentation of duality results for categories of coalgebras, and also lead to new duality theorems.
References
- [1] G. Bezhanishvili, N. Bezhanishvili, and J. Harding, Modal compact Hausdorff
spaces, Journal of Logic and Computation, 25 (2012), pp. 1–35.
- [2] P. T. Johnstone, Stone spaces, vol. 3 of Cambridge Studies in Advanced Mathematics,
Cambridge University Press, Cambridge, 1986. Reprint of the 1982 edition.
- [3] C. Kupke, A. Kurz, and Y. Venema, Stone coalgebras, Theoretical Computer Sci- ence, 327 (2004), pp. 109–134.
- [4] H.-E. Porst and W. Tholen, Concrete dualities, in Category theory at work, H. Her- rlich and H.-E. Porst, eds., vol. 18 of Res. Exp. Math., Heldermann Verlag, Berlin, 1991, pp. 111–136.