categories: Re: strict, strong, pseudo, lax

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

Keith Harbaugh summarized the terminology for "strictness":

> 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
>
[...] 
>
> Finally, in the USAGE EXAMPLES, for X take functor or morphism or
> (natural) transformation or algebra or ....

This is useful, but has a thread missing: what about when X is the
categorical structure itself?  E.g. for monoidal categories the most common
terminology is probably [I].  (There is a sensible notion of lax monoidal
category, even though it doesn't seem to have come up much.)  For
n-categories, however, the word "weak" comes in (at the same level as
"strong"!); some authors have also used "non-strict".  Traditionally
"n-category" on its own has meant the strict version, but there seems to be
gathering weight behind the opinion that it should mean the weak version.
Me, I've taken to using the system "strict/weak/lax" in most situations. 

I'm not really a fan of the word "weak": it's not very evocative somehow.
I thought that "fair" might be a good substitute.  It works well in the
metaphor of discipline conjured up by "strict" and "lax", sitting centrally
between the two extremes.  This usage is also sympathetic to the point of
view that up-to-iso is the default or natural level for things to be done
at: a "fair n-category" is what an n-category *ought* to be.  (You could
try "just" instead, but a "just n-category" sounds too much like an
only-just n-category...)  

Tom