symmetric monoidal (∞,1)-category of spectra
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monoidal dagger-category?
Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
A monoidal monad is a monad in the 2-category of monoidal categories, lax monoidal functors, and monoidal transformations.
The notion of monoidal monad is equivalent to the notion of commutative monad. To put it another way: to give a commutative strength for a monad is to give a monoidal monad whose underlying monad is . We explore this connection below.
As a preliminary, let be a monoidal category. We say a functor is strong if there are given left and right tensorial strengths
which are suitably compatible with one another. The full set of coherence conditions may be summarized by saying preserves the two-sided monoidal action of on itself, in an appropriate 2-categorical sense. More precisely: the two-sided action of on itself is a lax functor of 2-categories
( is the one-object 2-category associated with a monoidal category , and 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 .
In the setting where is symmetric monoidal, we will assume that the left and right strengths and are related by the symmetry in the obvious way, by a commutative square
where the ‘s are instances of the symmetry isomorphism.
There is a category of strong functors , where the morphisms are transformations which are compatible with the strengths in the obvious sense. Under composition, this is a strict monoidal category.
Monoids in this monoidal category are called strong monads.
A strong monad (def. ) is a commutative monad if there is an equality of natural transformations where
is the composite
is the composite
Let be a monoidal monad, with structural constraints on the underlying functor denoted by
Define strengths on both the left and the right by
is a commutative monad.
In fact, the two composites
are both equal to . We show this for the second composite; the proof is similar for the first. If denotes the monoidal constraint for and the constraint for the composite , then by definition is the composite given by
and so, using the properties of monoidal monads, we have a commutative diagram
which completes the proof.
The correspondence between monoidal monads and commutative monads is functorial. More precisely,
Given monoidal monads and on a monoidal category , a morphism of monads is a morphism of strong monads if and only if it is a monoidal natural transformation.
For a reference, see FPR ‘19, Proposition C.5.
See here.
The Kleisli category of a monoidal monad on inherits the monoidal structure from . In particular, the tensor product is given
where is the monoidal multiplication of . * The associator and unitor are induced by those of .
See examples of commutative monads.
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 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
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