Towards a Formal Theory of Graded Monads

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 \(\mathscr {C}\).

[1]  Paul-André Melliès.  Local States in String Diagrams , 2014, RTA-TLCA.

[2]  Paul-André Melliès,et al.  Segal Condition Meets Computational Effects , 2010, 2010 25th Annual IEEE Symposium on Logic in Computer Science.

[3]  Shin-ya Katsumata,et al.  Parametric effect monads and semantics of effect systems , 2014, POPL.

[4]  Claudio Hermida,et al.  Descent on 2-Fibrations and Strongly 2-Regular 2-Categories , 2004, Appl. Categorical Struct..

[5]  Ugo Dal Lago,et al.  Linear Dependent Types and Relative Completeness , 2011, 2011 IEEE 26th Annual Symposium on Logic in Computer Science.

[6]  Paul-André Melliès,et al.  Game Semantics in String Diagrams , 2012, 2012 27th Annual IEEE Symposium on Logic in Computer Science.

[7]  Patricia Johann,et al.  Generic Fibrational Induction , 2012, Log. Methods Comput. Sci..

[8]  Alan Mycroft,et al.  Coeffects: Unified Static Analysis of Context-Dependence , 2013, ICALP.

[9]  Stefan Milius,et al.  Generic Trace Semantics and Graded Monads , 2015, CALCO.

[10]  Paul-André Melliès,et al.  Finitary Semantics of Linear Logic and Higher-Order Model-Checking , 2015, MFCS.

[11]  S. Lack,et al.  The formal theory of monads II , 2002 .

[12]  Paul-André Melliès,et al.  Relational Semantics of Linear Logic and Higher-order Model Checking , 2015, CSL.

[13]  R. Street.  Two constructions on Lax functors , 1972 .

[14]  A. L. Smirnov.  Graded monads and rings of polynomials , 2008 .

[15]  J. Benabou.  Introduction to bicategories , 1967 .

[16]  Clemens Berger,et al.  Monads with arities and their associated theories , 2011, 1101.3064.

[17]  Gordon D. Plotkin,et al.  Notions of Computation Determine Monads , 2002, FoSSaCS.

[18]  Ugo Dal Lago.  A Short Introduction to Implicit Computational Complexity , 2010, ESSLLI.

[19]  Paul-André Melliès,et al.  The parametric continuation monad † , 2015, Mathematical Structures in Computer Science.