I had not yet sent out the following query of Stasheff. The response from the category list might be of interest to some on toplist...DMD ___________________________________________________________-- J. Stasheff wrote: >Has terminology settled down? >I can recall seeing various terms for >``simplicial object without degeneracies'' Date: Fri, 28 Jan 2000 07:01:42 -0500 (EST) From: James Stasheff Subject: Re: categories: terminology (fwd) So far I've had response mostly from the category side not from this list. Sor far the clear front runner is `face complex' Grandis, below, points out why the historical semi-simplicial won't fly for at least another generation. .oooO Jim Stasheff jds@math.unc.edu (UNC) Math-UNC (919)-962-9607 \ ( Chapel Hill NC FAX:(919)-962-2568 \*) 27599-3250 http://www.math.unc.edu/Faculty/jds ---------- Forwarded message ---------- Date: Fri, 28 Jan 2000 10:57:40 +0100 From: Marco Grandis To: categories@mta.ca, James Stasheff Subject: Re: categories: terminology I am afraid it has not. In my opinion, it should be called 'semi-simplicial object', consistently with the original terminology in Eilenberg-Zilber (see references below). Such a term has been adopted in Weibel's text on homological algebra (1994). But there seems to be some opposition. ___ I hope the following reconstruction of terminology is correct. 1. What is now called a simplicial object was introduced by Eilenberg and Zilber (1950); they use: (a) [already existing] 'simplicial complex' = set with distinguished parts; (b) [new term] 'semi-simplicial complex' = graded set with faces; (c) [new term] 'complete s.s. complex' = graded set with faces and degeneracies; 2. Later, notion (c) was recognised as more important than (b) and called 'semi-simplicial complex', leaving (b) without any standard name. 3. Since May's book (1967) at least, notion (c) gradually settled down as 'simplicial set', generalised to 'simplicial object' in a category; this is now standard. 4. It should now be natural to use a similar term, 'semi-simplicial object (possibly: set)' for (b), i.e. a 'simplicial object without degeneracies' (as in Weibel 1994). This is consistent with the original use in Eilenberg-Zilber and gives a non-ambiguous set of terms for the three notions recalled: (a) 'simplicial complex' (also: combinatorial complex) (b) 'semi-simplicial object (set)' (c) 'simplicial object (set)' However, I used myself this terminology in a paper published in '97 and had strong reactions from people attached to the terminology in use between 50's and '60s (point 2 above). ___ References: S. Eilenberg - J.A. Zilber, Semi-simplicial complexes and singular homology, Ann. of Math. 51 (1950), 499-513. J.P. May, Simplicial objects in algebraic topology, Van Nostrand 1967. C.A. Weibel, An introduction to homological algebra, Cambridge Univ. Press, Cambridge, 1994. ___ With best regards Marco Grandis Dipartimento di Matematica Universita' di Genova via Dodecaneso 35 16146 GENOVA, Italy e-mail: grandis@dima.unige.it tel: +39.010.353 6805 fax: +39.010.353 6752 http://www.dima.unige.it/STAFF/GRANDIS/ ftp://pitagora.dima.unige.it/WWW/FTP/GRANDIS/