categories: strict, strong, pseudo, lax

["Keith Harbaugh" <harbaugh_keith@hotmail.com>]

David Benson on 2002-02-08 wrote:

  Early on, John Gray and others noticed that functors ought not
  compose on-the-nose, but up-to-isomorphism, and (I believe)
  called the result <<pseudo-functors>>.  Well, there is nothing
  <<pseudo>> about it and it is these latter entities that deserve
  the name <<functor>>, with the first 50 years of use then becoming
       XXX functor
  where XXX might be <<young>> or <<nasal>> or ...

The issues here seem to be:
What are the possible comparison methods for composition, identity, etc.,
how shall they be briefly named,
and which shall be the default?

The most conventional answers to these questions seem to be:

. COMPARISON  |         SYSTEMS        |        USAGE EXAMPLES
.             |   [E]     [I]     [M]  |   [E]       [I]       [M]
. ___________ | ______  ______  ______ | ________  ________  ________
. equality    | ------  strict  strict | X         strict X  strict X
. isomorphism | pseudo  ------  strong | pseudo-X  X         strong X
. morphism    | lax     lax     ------ | lax X     lax X     X

The leftmost column gives the three most common comparison methods.
(morphism actually has two variants depending on the sense of the morphism.)
Next come three different naming systems for the three comparison methods.
[E], [I] and [M] designate both a system and a reference for it:

[E] Kelly and Street, Review of the Elements of 2-Categories (1974), Section 
1.5
[I] Blackwell, Kelly and Power, Two-Dimensional Monad Theory (1989), Section 
1.2
[M] same reference as [E];

each reference provides examples, historical justification,
and comparisons to alternative conventions
(e.g., in the setting of bicategories, homomorphism for strong morphism).
In each naming system the default comparison method has a '------'
(the lack of a name) as its "name".
Finally, in the USAGE EXAMPLES, for X take functor or morphism or
(natural) transformation or algebra or ....

The original quotation suggested taking up-to-isomorphism as the norm;
that would seem to put it squarely in the [I] camp, making its XXX = 
<<strict>>.

It would be interesting to expand the table to take into account
other comparison methods and systems;
also it would be in some ways be simpler
if the isomorphism comparison could be denoted by a single term,
but pseudo and strong have the advantage of clearly indicating
where they stand relative to the norms in their respective systems.

It has been stated that the notion of pseudo-functor originated with
Grothendieck's study of fibrations (fibred categories) in the 1961 SGA
Cat\'egories fibr\'ees et descente;
perhaps Madame Ehreshmann or Monsieur Benabou
might care to expand, confirm or comment on that...

Keith