Algebras and modules
Model category presentations
Geometry on formal duals of algebras
With duals for objects
With duals for morphisms
Special sorts of products
In higher category theory
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
Tensorial strengths and commutative monads
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 .
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. 2) is a commutative monad if there is an equality of natural transformations where
is the composite
is the composite
From monoidal monads to commutative monads
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 first composite; the proof is similar for the second. 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.