Subject: categories: Re: terminology

From: [email-masked] (Marco Grandis)
Date: Fri, 28 Jan 2000 10:57:40 +0100

J. Stasheff wrote:

>Has terminology settled down?
>I can recall seeing various terms for
>``simplicial object without degeneracies''

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.
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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).

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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.

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With best regards

Marco Grandis

Dipartimento di Matematica
Universita' di Genova
via Dodecaneso 35
16146 GENOVA, Italy

e-mail: [email-masked]
tel: +39.010.353 6805   fax: +39.010.353 6752

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