Towards a Formal Theory of Graded Monads

  • Soichiro Fujii
  • Shin-ya Katsumata
  • Paul-André Melliès
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9634)


We initiate a formal theory of graded monads whose purpose is to adapt and to extend the formal theory of monads developed by Street in the early 1970’s. We establish in particular that every graded monad can be factored in two different ways as a strict action transported along a left adjoint functor. We also explain in what sense the first construction generalizes the Eilenberg-Moore construction while the second construction generalizes the Kleisli construction. Finally, we illustrate the Eilenberg-Moore construction on the graded state monad induced by any object V in a symmetric monoidal closed category \(\mathscr {C}\).


Monoidal Category Tensorial Logic Object Versus Cartesian Closed Category Kleisli Category 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The authors are grateful to the anonymous reviewer for suggesting an alternative and more elegant construction of the graded state monad. The authors are also grateful to Marco Gaboardi and to Dominic Orchard for a number of useful discussions about this work. The authors were supported by the JSPS-INRIA Bilateral Joint Research Project CRECOGI, the second author was supported by Grant-in-Aid No.15K00014 while the third author was partly supported by the ANR Project Recre.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Soichiro Fujii
    • 1
  • Shin-ya Katsumata
    • 2
  • Paul-André Melliès
    • 3
  1. 1.Department of Computer ScienceThe University of TokyoTokyoJapan
  2. 2.RIMSKyoto UniversityKyotoJapan
  3. 3.Laboratoire PPS, CNRSUniv. Paris DiderotParisFrance

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