Weak monoids and monads in compact bicategories

[Mike Stay <metaweta@gmail.com>]

I'm looking for references to who first published these various
algebra-like and category-like constructions, all either weak monoids
or monads in compact bicategories:

- If C has finite products and pullbacks, a weak monoid in Span(C) is
a categorification of an associative algebra, while a monad is a
category internal to C.  (I think Benabou pointed out the latter.)

- In Rel, a weak monoid is a Boolean algebra, while a monad is a preorder.

- A 2-rig is a symmetric monoidal category where the tensor product
distributes over the colimits.  Given a 2-rig R, Mat(R) is the
bicategory of natural numbers, matrices of objects of R, and matrices
of morphisms of R.  A weak monoid in Mat(R) is a categorified
finite-dimensional associative algebra, something like a finite field.
  A monad in Mat(R) is a finite R-enriched category.

- A weak monoid in Prof is a promonoidal category.  A symmetric
monoidal monad in Prof is an "Arrow" in the sense of Hughes and is
related to Freyd categories.  Is there a name for an arbitrary monad
in Prof other than "monad in Prof"?


-- 
Mike Stay - metaweta@gmail.com
http://www.cs.auckland.ac.nz/~mike
http://reperiendi.wordpress.com


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