Graded Algebraic Theories [chapter]

Satoshi Kura
2020 Lecture Notes in Computer Science  
We provide graded extensions of algebraic theories and Lawvere theories that correspond to graded monads. We prove that graded algebraic theories, graded Lawvere theories, and finitary graded monads are equivalent via equivalence of categories, which extends the equivalence for monads. We also give sums and tensor products of graded algebraic theories to combine computational effects as an example of importing techniques based on algebraic theories to graded monads. S. Kura Proof. By the
more » ... ence [Set, Set] f [ℵ 0 , Set] induced by restriction and the left Kan extension along the inclusion i : ℵ 0 → Set. Day Convolution We describe a monoidal biclosed structure on the ( covariant) presheaf category [M, Set] 0 where M = (M, ⊗, I) is a small monoidal category [3]. Here, we use the subscript (−) 0 to indicate that [M, Set] 0 is an ordinary (not enriched) category since we also use the enriched version [M, Set] later. The external tensor product F G : M × M → Set is defined by (F G)(m 1 , m 2 ) = F m 1 × Gm 2 for any F, G : M → Set. Definition 4. Let F, G : M → Set be functors. The Day tensor product F⊗ G : M → Set is the left Kan extension Lan ⊗ (F G) of the external tensor product F G : M × M → Set along the tensor product ⊗ : M × M → M. Note that a natural transformation θ : F⊗ G → H is equivalent to a natural transformation θ m1,m2 : F m 1 × Gm 2 → H(m 1 ⊗ m 2 ) by the universal property. The Day convolution induces a monoidal biclosed structure in [M, Set] 0 [3]. Proposition 5. The Day tensor product makes ([M, Set] 0 ,⊗, y(I)) a monoidal biclosed category where y : M op → [M, Set] 0 is the Yoneda embedding y(m) := M(m, −). The left and the right closed structure are given by F, G m = [M, Set] 0 (F, G(m⊗−)) and [F, G] m = [M, Set] 0 (F, G(−⊗m)) for each m ∈ M, respectively. Note that since we do not assume M to be symmetric, neither is [M, Set] 0 . Note also that the twisting and the above construction commute: there is an isomorphism [M, Set] 0 t ∼ = [M t , Set] 0 of monoidal categories. 404 S. Kura
doi:10.1007/978-3-030-45231-5_21 fatcat:e26xprsfgvey5ktqn6xduav6em