bar construction for better-than-average monads

bar construction for better-than-average monads

[John Baez]

Hi -

Given a monad T: C -> C, any algebra A of this monad gives rise to a
simplicial algebra of this monad, say BA, via the bar construction.

It's finally occurred to me to wonder: how do various extra structures on
the monad give extra structures on the bar construction?

For example, suppose C is monoidal.  Then we say T is a "strong" monad if
it's a monad not just in Cat but in the 2-category of C-actions.  This
amounts to having natural transformations

A tensor TB -> T(A tensor B)

obeying various laws.  If T is strong, what does this do for its bar
construction?

Similarly, we say T is a "monoidal" monad if it's a monad not just in Cat
but in the 2-category of monoidal categories, lax monoidal functors and
monoidal natural transformations.  This amounts to having a natural
transformation

TA tensor TB -> T(A tensor B)

and a morphism I -> TI obeying various laws.  If T is monoidal, what does
this do for its bar construction?

I could ask the same question for commutative monads, or for a pair of
monads related by a distributive law.

I would enjoy figuring these out, but someone must have done it already,
and I'd rather enjoy doing something new.

Best,
jb


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