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Diads and their Application to Topoi


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.

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Corresponding author

Correspondence to Toby Kenney.

Additional information

Funded by an AARMS postdoctoral fellowship. I would also like to thank Geoff Cruttwell, Robert Paré, and Richard Wood for a lot of helpful discussions.

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Kenney, T. Diads and their Application to Topoi. Appl Categor Struct 17, 567 (2009).

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  • Category
  • Diad
  • Dialgebras
  • Topos
  • Comonad
  • Idempotent monad

Mathematics Subject Classifications (2000)

  • 18A40
  • 18B25