Re: the walking adjunction and biadjunction

Re: the walking adjunction and biadjunction

[Tom Leinster]

Toby Bartels wrote:
> 
> For example, one could define the idea of multiplication in a monoid
> as a binary operation and a nullary operation
> or alternatively as an operation on finite tuples.
> The former is more common, but I prefer the latter;
> who has the right idea?

An interesting question in itself.  I don't think either idea is "right", but
I (presumably) share with you the feeling that often the latter is more
appropriate.  However, if you resolve wholeheartedly never to use a binary +
nullary presentation of a monoid-like structure then you actually find
yourself in quite an extreme position.  For instance, a monoid would be
defined as a set M together with an n-fold operation

(m_1, ..., m_n) |---> [m_1 ... m_n]

on M for each natural n, subject to axioms.  This is as expected so far, but
we've disallowed ourselves from using what would probably be the natural
choice of axioms,

[[m_1^1 ... m_1^{k_1}] ... [m_n^1 ... m_n^{k_n}]] = [m_1^1 ... m_n^{k_n}],
m = [m],

since this is a binary + nullary presentation.  So instead we should derive
from the n-fold multiplications a k-ary operation o_T on M for each (finite,
planar) k-leafed tree T; and the axioms then become that o_T = o_U for any
two k-leafed trees T and U.

The situation gets more extreme still if you want a wholeheartedly
non-binary-and-nullary presentation of the notion of monoidal category.  We
have an underlying category M, an n-fold tensor functor for each n, and then
coherence cells obeying coherence axioms.  The obvious choice for the
coherence cells comes from turning the two equations above into specified
isomorphisms, but again this is disallowed, so we have to specify a coherence
cell o_T --~--> o_U for each T and U, where o_T, o_U are now derived tensor
functors.  Then we need to put axioms on the coherence cells, and once more
the obvious way of doing this involves something of a binary + nullary
character.  Specifically, you have to make sure that the coherence cells o_T
--~--> o_U are compatible with "grafting of trees", which means taking a
k-leafed tree T and sticking onto its leaves k trees T_1, ..., T_k, to make a
new tree T(T_1, ..., T_k) - but this expression has *2* (bad number!) levels
of trees.  So we need to replace these axioms with equivalent
non-binary-and-nullary ones, and this means considering more complicated
structures still.

(The considerations in the last paragraph are really to do with writing down
a non-binary-and-nullary presentation of the theory of operads, which are
themselves monoids of a sort.)

Tom

Sketches and ideas (Was: the walking adjunction and biadjunction)

[Toby Bartels]

Tom Leinster wrote in part:

>Toby Bartels wrote:

>>For example, one could define the idea of multiplication in a monoid
>>as a binary operation and a nullary operation
>>or alternatively as an operation on finite tuples.
>>The former is more common, but I prefer the latter;
>>who has the right idea?

>An interesting question in itself.  I don't think either idea is "right",

That was supposed to be my point.
Just as a group can be described many ways by generators and relations,
so a sketch (if we define a sketch to be an entire category;
apparently that varies) can be described many ways by ideas.
It's the category that truly characterises what a monoid is
(in the given doctrine), so it better deserves the name "idea",
if we're trying to hark back to Plato-n (even just to be cute).
(Whether or not it's too late to change, I can't say.)

>we've disallowed ourselves from using what would probably be the natural
>choice of axioms,

>[[m_1^1 ... m_1^{k_1}] ... [m_n^1 ... m_n^{k_n}]] = [m_1^1 ... m_n^{k_n}],
>m = [m],

>since this is a binary + nullary presentation.

2 indices and 0 indices.  *Gulp*  You're right!
I always felt annoyed having to write in that nullary axiom; now I know why.

>So instead we should derive
>from the n-fold multiplications a k-ary operation o_T on M for each (finite,
>planar) k-leafed tree T; and the axioms then become that o_T = o_U for any
>two k-leafed trees T and U.

I suppose that you're aware of this, but note that we need to allow
nodes that don't branch but also aren't labelled (considered leaves),
which is where we place [].  For example, the tree

m           .
 \         /
   \     /
     \./

indicates the product [m[]]; that it equals m is the right unit law.
This threw me for a moment, since [] seemed at first to have disappeared.

We could also go straight to trees and define them as the basic operations,
then requiring as axiom that grafting of trees produces the same result
as composing the operations.

If we were defining a nonassociative operation without identity,
then we could denote the basic operations by *binary* trees.

>Specifically, you have to make sure that the coherence cells o_T
>--~--> o_U are compatible with "grafting of trees", which means taking a
>k-leafed tree T and sticking onto its leaves k trees T_1, ..., T_k, to make a
>new tree T(T_1, ..., T_k) - but this expression has *2* (bad number!) levels
>of trees.

The nullary counterpart is grafting 0 trees to get the tree m.

>So we need to replace these axioms with equivalent
>non-binary-and-nullary ones, and this means considering more complicated
>structures still.

Well, I managed to introduce grafting back before the categorification!
Aren't I clever?  Too clever for my own good?  ^_^

>(The considerations in the last paragraph are really to do with writing down
>a non-binary-and-nullary presentation of the theory of operads, which are
>themselves monoids of a sort.)

As long as we learn the lesson that binary operations
warrant a search for their nullary partners,
then we've done the important thing, at least,
even if we miss out on some ever more complicated elegance.


-- Toby
   toby@math.ucr.edu