Towards a Formal Theory of Graded Monads

Soichiro Fujii; Shin-ya Katsumata; Paul-André Melliès

doi:10.1007/978-3-662-49630-5_30

International Conference on Foundations of Software Science and Computation Structures

FoSSaCS 2016: Foundations of Software Science and Computation Structures pp 513-530 | Cite as

Towards a Formal Theory of Graded Monads

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Soichiro Fujii
Shin-ya Katsumata
Paul-André Melliès

Conference paper

Part of the Lecture Notes in Computer Science book series (LNCS, volume 9634)

Abstract

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 $\mathscr {C}$.

Keywords

Monoidal Category Tensorial Logic Object Versus Cartesian Closed Category Kleisli Category

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Notes

Acknowledgments

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.

Appendix: An Algebraic Presentation of Graded $\mathbf {S}$-Algebras

Formulated in the language of multisorted universal algebra, a graded $\mathbf {S}$-algebra (A, h) is completely determined by a family $(A_m)_{m\in \mathbf{N}}$ of sets equipped with an operation $A_v:A_m\rightarrow A_n$ for each injection $v:m\rightarrow n$ and with an operation $\langle f,n \rangle : A_n^{V^m}\rightarrow A_{m+n}$ for each pair of natural numbers m, n and each function $f:V^m\rightarrow V^m$. These operations should moreover satisfy the equations below. First of all, the two functoriality equations below:

$$ A_{\mathrm {id}_{m}} = \mathrm {id}_{A_m} \qquad \qquad \qquad A_{w\circ v}= A_w\circ A_v. $$

Then, the naturality equation, for all injections $v:m\rightarrow m^\prime $ and $w:n\rightarrow n^\prime $,

$$ A_{v+w}(\langle f,n \rangle (a_i)_{i\in V^m})\quad =\quad \langle v\bullet f,n^\prime \rangle (A_w(a_{i^\prime \circ v}))_{i^\prime \in V^{m^\prime }}, $$

where the function $v\bullet f:V^{m^\prime }\rightarrow V^{m^\prime }$ is defined as

$$ \big ((v\bullet f) (val_{k^\prime })_{k^\prime \in m^\prime }\big )_{l^\prime } \quad =\quad {\left\{ \begin{array}{ll} \big (f(val_{v(k)})_{k\in m}\big )_l &{}\text{ if } l^\prime = v(l) \text{ for } \text{ some } l\in m;\\ val_{l^\prime } &{} \text{ otherwise. }\end{array}\right. } $$

Then, the unit equation

$$ \langle \mathrm {id}_{V^0},m \rangle \quad =\quad \mathrm {id}_{A_m}, $$

for a function $f:V^m\rightarrow V^m$ and an injection $v:m\rightarrow m^\prime $ ; and finally the associativity equation

$$ \langle f \rangle _{n+p}(\langle g_i,p \rangle (a_{i,j})_{j\in V^n})_{i\in V^m} \quad = \quad \langle f\star (g_i)_{i\in V^m}, p \rangle (a_{i,j})_{i,j\in V^{m+n}}, $$

where the function

$$f\star (g_i)_{i\in V^m} \quad : \quad V^{m+n} \quad \longrightarrow \quad V^{m+n}$$

is defined as follows, for a function $f:V^m\rightarrow V^m$ and for a family of functions $(g_i:V^n\rightarrow V^n)_{i\in V^m}$:

$$ \big ((f\star (g_i)_{i\in V^m}) (val_{k},wal_{k^\prime })_{k\in m,k^\prime \in n}\big )_{l} ={\left\{ \begin{array}{ll} \big (f(val_k)_{k\in m}\big )_l &{}\text{ if } l\in m,\\ \big (g_{(val_k)_{k\in m}}(wal_{k^\prime })_{k^\prime \in n}\big )_l &{} \text{ if } l\in n. \end{array}\right. } $$

Homomorphisms of graded $\mathbf {S}$-algebras are just families of maps $\varphi _m:A_m\rightarrow A^\prime _m$ commuting with the operations introduced above.

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Authors and Affiliations

Soichiro Fujii
- 1
Shin-ya Katsumata
- 2
Email author
Paul-André Melliès
- 3

1.Department of Computer ScienceThe University of TokyoTokyoJapan
2.RIMSKyoto UniversityKyotoJapan
3.Laboratoire PPS, CNRSUniv. Paris DiderotParisFrance

Towards a Formal Theory of Graded Monads

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