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 21-Feb-2002 23:50:20 -0400,1854;000000000001-00000000