Monad for weakly monoidal categories
We’ve got free and forgetful functors Define
Given a category
the category
has
- binary trees with
labeled leaves as objects and
- binary trees with
labeled leaves together with the natural isomorphisms from the definition of a weakly monoidal category as its morphisms.
The multiplication collapses two layers of trees down to one. The unit
gives a one-leaf tree.
An algebra of the monad is a category together with a functor
such that
and
Define
Then the associator should be a morphism
However, it isn’t immediately evident that the associator that comes from does the job, since just applying
to
gives
for the source instead of
,
which we get by replacing with its definition above. We need an isomorphism
so we can define Now we use the equations an algebra has to satisfy to derive this isomorphism. Since
the following two objects are equal:
Therefore, the isomorphism we wanted is simply equality and
It also means that
satisfies the pentagon equation.
A similar derivation works for the unitors and the triangle equation.
A morphism of algebras is a functor such that
Now
and
so we have the coherence laws for a strict monoidal functor.
Also,
so it preserves the associator as well. The unitors follow in the same way, so morphisms of these algebras are strict monoidal functors that preserve the associator and unitors.
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