Diads and their Application to Topoi

  • Toby KenneyEmail author


It is well known that the category of coalgebras for a finite-limit preserving comonad on a topos is again a topos, and the category of algebras for a finite-limit preserving monad is a topos if the monad is idempotent, but not in general. A generalisation of this result (Paré et al., Bull Aus Math Soc 39(3):421–431, 1989) is that the full subcategory of fixed points for any idempotent finite-limit preserving endofunctor is again a topos (and indeed a subquotient in the category of topoi and geometric morphisms). Here, we present a common generalisation of all the above results, based on a notion which we call a diad, which is a common generalisation of a monad and a comonad. Many of the constructions that can be applied to monads and comonads can be extended to all diads. In particular, the category of algebras or coalgebras can be generalised to a category of dialgebras for a diad. The generalisation we present here is that the category of dialgebras for a finite-limit preserving left diad (for example, the diad corresponding to a comonad, or any idempotent endofunctor) on a topos is again a topos.


Category Diad Dialgebras Topos Comonad Idempotent monad 

Mathematics Subject Classifications (2000)

18A40 18B25 


  1. 1.
    Johnstone, P.T.: Sketches of an Elephant: A Topos Theory Compendium, volume 1 and 2 of Oxford Logic Guides. Clarendon Press (2002)Google Scholar
  2. 2.
    Kenney, T.: General Theory of Diads. Appl. Categ. Structures (2008, submitted)Google Scholar
  3. 3.
    Koslowski, J.: Monads and interpolads in bicategories. Theory Appl. Categ. 3(8), 182–212 (1997)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Paré, R., Rosebrugh, R., Wood, R.J.: Idempotents in bicategories. Bull. Austral. Math. Soc. 39(3), 421–434 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Riguet, J.: Relations binaires, etc. Bull. Soc. Math. France 76, 114–155 (1948)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Riguet, J.: Quelques propriétés des relations difonctionelles. C. R. Acad. Sci. Paris 230, 1999–2000 (1950)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Dalhousie UniversityHalifaxCanada

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