categories: Re: Time for functors to grow up; three queries

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

I wanted to respond to just a very small part of David Benson's interesting
post:

> Early on, John Gray and others noticed that functors ought not
> compose on-the-nose, but up-to-isomorphism

(Presumably this is referring to ordinary functors between ordinary
categories, and "compose on-the-nose" means that composition of functors
obeys associativity and identity axioms on the nose, not just up to natural
iso.)

I've been thinking about this phenomenon of "unreasonable strictness"
recently, and am in two minds about David's "ought".  One mind says
"functors *do* compose on the nose, and if our intuition expects that they
ought not to then our intuition needs refining".  The other says "maybe the
fact that they do compose on the nose is just an artefact of our particular
choice of formal system, and maybe there's some more righteous formal
system in which they don't compose on the nose".  At present I'm tending
towards the former view: that there's something interesting going on with
unexpected strictnesses, that we don't yet understand.  Here are some
examples of what I mean.

EQUIVALENCES.  When I've tried to teach the notion of equivalence of
categories to newcomers to the subject, I've said something like this.
In general we don't care whether two objects of a category are equal, only
whether they're isomorphic; for instance, if a group theorist says that two
groups are the same, she doesn't usually know or care whether, by some
miracle of set theory, they're actually equal - isomorphism is all that
matters.  So the notion of isomorphism of categories is unreasonably
strict, requiring as it does that certain equalities of objects occur.  We
therefore replace those equalities by (natural) isomorphisms to obtain the
notion of equivalence.  

All well and good and perfectly standard: but there follows an
uncomfortable moment.  With the build-up above, I've led my student to
expect that equality of objects and isomorphism of categories are such
outrageously over-strict notions that they hardly ever occur in nature;
that if you pluck out of the air a random example of two categories which
are "essentially the same" then that'll be an equivalence, not an
isomorphism.  Actually, that's not the case: there are plenty of perfectly
natural examples of isomorphic categories.  For instance, if you take two
different definitions of topological space - say, a set equipped with open
subsets or a set equipped with a closure operator, and the corresponding
notions of continuous map - then the resulting two categories are
isomorphic.  The same goes for different definitions of group obtained by
varying the generators and relations presenting the theory.  And the same
goes in other categories of sets-with-structure.  Moreover, any equivalence
between one-object categories (monoids) is in fact an isomorphism, being
full and faithful.  So isomorphism of categories seems not to be as
unreasonable a notion as the standard rationale would have you believe.

MONADICITY.  The situation's even more extreme here.  Right now I can't
think of *any* natural example of a monadic adjunction in which the
comparison functor is an equivalence but not an isomorphism (though I'm
sure someone can put me right).  Certainly it's an isomorphism for the
motivating family of examples of monadic adjunctions: algebraic theories on
Set.  And this "unreasonable strictness" leads into...

CREATION.  Let's say that a functor D ---> C *creates limits strictly* if
given any diagram in D and limit cone on the corresponding diagram in C,
the limit cone lifts uniquely to a cone on the diagram in D, and moreover
that cone in D is a limit cone.  This is, of course, a "bad" definition,
informally because it refers to equality of objects, and formally because
an equivalence of categories needn't create limits strictly.  The "good"
definition of creation relaxes these equalities to isomorphisms.  

However, it's undeniable that the strict notion is useful (as well as being
easier to teach).  For example, all the usual monadic functors (referred to
above) create limits and coequalizers for such-and-such pairs *strictly*.
So does the forgetful functor Set^C ---> Set^{C_0}, where C is a category
and C_0 its underlying set.  So, indeed, does the forgetful functor C^T
---> C coming from a monad T on a category C.

(Of course, the fact that creation of such-and-such coequalizers is strict
corresponds precisely to the comparison functor actually being an
isomorphism.  This is the level at which the Monadicity Theorem is proved
in Categories for the Working Mathematician, but these days that seems
immoral.)

DEFINITIONS OF BICATEGORY.  So far my examples have been of situations
where something happens one degree more strictly than it "ought" to.  Here's
one where it happens two degrees "too" strictly.  

In the usual definition of bicategory, you have a binary composition of
hom-categories and an identity cell on each object.  Put another way, this
is k-fold composition for k=2 and k=0.  You can redefine "bicategory" by
specifying instead a k-fold composition for all natural numbers k, adding,
of course, suitable coherence cells obeying suitable coherence axioms.

Now, you want to say that the new definition of bicategory is essentially
the same as the old one.  The structure formed by bicategories is
3-dimensional: you have bicategories, weak functors, weak transformations,
and modifications.  So in order to state the equivalence between the two
definitions of bicategories, you "ought" to have to make a statement
involving 3-dimensional structures: specifically, that the two different
*tricategories* of bicategories are *triequivalent*.  

In fact, the comparison can be made at the level of (1-)categories: with a
suitable notion of weak functor between bicategories-in-the-new-sense, the
two *categories* of bicategories are *equivalent*.  It's not hard to see
why this happens: essentially it's because when you start with a bicategory
in one definition and create from it a bicategory in the other definition,
you're not changing the underlying graph of 0-, 1- and 2-cells.

Finally, the fact that David mentioned (if I interpret him correctly) can
be phrased as "Cat is a (strict) 2-category, not just a bicategory".  This
is exactly what makes the Yoneda proof of the coherence theorem for
bicategories work (namely, if B is a bicategory then B embeds nicely into
the 2-category [B^op,Cat] and is thus equivalent to a 2-category).  So if
we regard it as accidental that functors compose on the nose then we should
also regard it as accidental that every bicategory is equivalent to a
2-category - not necessarily an unreasonable point of view, but again it
seems to be ignoring the fine grain.

Sorry to go on for so long.  I feel that there is some refinement to be
made to our intuition about strictness and weakness, but I don't know what
that refinement should be.

Tom