monoidal monad (Rev #18, changes) in nLab

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Definition

Tensorial strengths and commutative monads

As a preliminary, let VV be a monoidal category. We say a functor T:V→VT \colon V \to V is strong if there are given left and right tensorial strengths

τ A,B:A⊗T(B)→T(A⊗B)\tau_{A, B} \colon A \otimes T(B) \to T(A \otimes B)

\,

σ A,B:T(A)⊗B→T(A⊗B).\sigma_{A, B} \colon T(A) \otimes B \to T(A \otimes B).

which are suitably compatible with one another. The full set of coherence conditions may be summarized by saying TT preserves the two-sided monoidal action of VV on itself, in an appropriate 2-categorical sense. More precisely: the two-sided action of VV on itself is a lax functor of 2-categories

V˜:BV×(BV) op→Cat\tilde{V} \colon B V \times (B V)^{op} \to Cat

(BVB V is the one-object 2-category associated with a monoidal category VV, and (BV) op(B V)^{op} is the same 2-category but with 1-cell composition (= tensoring) in reverse order), and the two-sided strength means we have a structure of lax natural transformation V˜→V˜\tilde{V} \to \tilde{V}.

There is a category of strong functors V→VV \to V, where the morphisms are transformations λ:S→T\lambda \colon S \to T which are compatible with the strengths in the obvious sense. Under composition, this is a strict monoidal category.

From monoidal monads to commutative monads

Let (T:V→V,u:1→T,m:TT→T)(T \colon V \to V, u \colon 1 \to T, m \colon T T \to T) be a monoidal monad, with structural constraints on the underlying functor denoted by

α A,B:T(A)⊗T(B)→T(A⊗B),ι=uI:I→T(I).\alpha_{A, B} \colon T(A) \otimes T(B) \to T(A \otimes B), \qquad \iota = u I: I \to T(I).

Define strengths on both the left and the right by

τ A,B=(A⊗T(B)→uA⊗1T(A)⊗T(B)→α A,BT(A⊗B)),\tau_{A, B} = (A \otimes T(B) \stackrel{u A \otimes 1}{\to} T(A) \otimes T(B) \stackrel{\alpha_{A, B}}{\to} T(A \otimes B)),

\,

σ A,B=(T(A)⊗B→1⊗uBT(A)⊗T(B)→α A,BT(A⊗B)).\sigma_{A, B} = (T(A) \otimes B \stackrel{1 \otimes u B}{\to} T(A) \otimes T(B) \stackrel{\alpha_{A, B}}{\to} T(A \otimes B)).

Functoriality of the correspondence

The correspondence between monoidal monads and commutative monads is functorial. More precisely,

For a reference, see FPR ‘19, Proposition C.5.

Tensor product of algebras and multimorphisms

See here.

Monoidal structure on the Kleisli category

The Kleisli category of a monoidal monad TT on CC inherits the monoidal structure from CC. In particular, the tensor product is given

• On objects, by the tensor product ⊗\otimes of CC;
• On morphisms, given k:X→TAk:X\to TA and h:Y→TBh:Y\to TB, their product is the map X⊗Y→T(A⊗B)X\otimes Y \to T(A\otimes B) obtained by the composition \begin{tikzcd} X\otimes Y \ar{r}{f\otimes g} & TA \otimes TB \ar{r}{\nabla} & T(A\otimes B), \end{tikzcd} where ∇\nabla is the monoidal multiplication of TT.
• The associator and unitor are induced by those of CC.

Examples

See examples of commutative monads.

See also

• strong monad, tensorial strength
• monoidal functor
• commutative monad, commutative algebraic theory

References

• Anders Kock, Monads on symmetric monoidal closed categories, Arch. Math. 21 (1970), 1–10.

• Anders Kock, Strong functors and monoidal monads, Arhus Universitet, Various Publications Series No. 11 (1970). PDF.

• Anders Kock, Closed categories generated by commutative monads (pdf)

• H. Lindner, Commutative monads in Deuxiéme colloque sur l’algébre des catégoriesDeuxiéme colloque sur l’algébre des catégories. Amiens-1975. Résumés des conférences, pages 283-288. Cahiers de topologie et géométrie différentielle catégoriques, tome 16, nr. 3, 1975.

• William Keigher, Symmetric monoidal closed categories generated by commutative adjoint monads, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 19 no. 3 (1978), p. 269-293 (NUMDAM, pdf)

• Gavin J. Seal, Tensors, monads and actions (arXiv:1205.0101)

• Martin Brandenburg, Tensor categorical foundations of algebraic geometry (arXiv:1410.1716)

A statement in the text appears in Appendix C of

• Tobias Fritz, Paolo Perrone and Sharwin Rezagholi, Probability, valuations, hyperspace: Three monads on Top and the support as a morphism, 2019 (arXiv:1910.03752)