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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]