Straw man terminology

[Peter May <may@math.uchicago.edu>]

As a small point, I'm quite sure Boardman and Vogt never used the
term ``weak Kan complex'' that Andre rightly opposes: they referred
to the ``restricted Kan condition'' (see SLN 347, p. 102). I have no
idea who is guilty of ``weak'' in this context, but I think it was never
in common use. In fact, Boardman and Vogt were right away sensitive
to size issues, referring to ``simplicial classes'' rather than
simplicial sets
satisfying the inner horn condition. They did not think of them as special
kinds of Kan complexes, which arguably they are not since Kan complexes
are generally understood to be simplicial sets.  Certainly that was how
Kan complexes were understood when Boardman and Vogt were writing
(published 1973, mostly written earlier). They already knew then that their
notion gave them, in their words, ``good substitutes for categories''.

I also like Andre's term ``quasi-categories'' and prefer it to the
infinity alternatives, for the reasons he gives. Like Kan complex,
it has a fixed and evocative definite meaning, unlikely to be confused
with anything else. Now if only he would publish ...

Responding to another part of this (endless) string, I also regret how
words with a nice ring to them get so modified that they no longer have
a clear precise meaning attached to them. Everyone can guess  one word
I'm thinking of (my mother was an opera singer).

Peter May




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