Re: the walking adjunction and biadjunction

[Tom Leinster <T.Leinster@dpmms.cam.ac.uk>]

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