* Weak monoids and monads in compact bicategories
@ 2013-07-10 5:44 Mike Stay
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From: Mike Stay @ 2013-07-10 5:44 UTC (permalink / raw)
To: categories
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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