categories: Linear Monads (Was Re: Tangle, Braid... related category?)

["Robert A.G. Seely" <rags@math.mcgill.ca>]

> Jules Bean <jules@jellybean.co.uk> wrote:
> Related to these two these is a category whose objects are again the
> natural numbers, and whose morphisms are pieces of string which are
> allowed to split into multiple strands, and join together into single
> strands, such as the following morphism 3 --> 2:
>
>  *   *   *
>   \ /   /
>    |   /\
>    \  /  |
>     \/   |
>      *   *

As Tom Leinster, Joachim Kock, and John Baez point out, there are a
number of studies of such structures.  But for a coherent general
perspective, I would also point out that this is what we (Robin
Cockett, J"urgen Koslowski, and Robert Seely) call a "linear monad" in
a linear bicategory ("Introduction to Linear Bicategories", Math
Struct in Comp Sci 10 (2000) 165--203, also available from my web
site, url as given below).  The general theory is worked out in
section 4 of that paper.  A linear bicategory may be thought of as a
bicategory with two (usually distinct) horizontal composition
operations (we think of them as tensors).  In figures such as above,
we can imagine one tensor as "tieing together" the top wires, and the
other for the bottom wires.  Monads and comonads may be defined in
this setting, and a linear monad is a pair consisting of a monad and a
comonad which are "compatible" with one another, each of which acts
(or coacts, as appropriate) upon the other.  As we point out in the
paper, these are a natural generalization of Frobenius algebras, to
pick up Joachim Kock's reference.

 - all the best, Robert (Seely)

==================
R.A.G. Seely
<rags@math.mcgill.ca>
<http://www.math.mcgill.ca/rags>