Talk:Simplicial set

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The first definition of simplicial set is wrong -- it seems to be the definition of cosimplicial set. The categorical definition is correct, and it's preferable, in any case. The first definition, via explicit formulas, is too verbose to be useful to a beginner, and could just be eliminated.

The example called "The standard simplicial set" is actually a cosimplicial set. It would more properly be called the standard cosimplicial space or the standard cosimplicial simplex.

As you can see, I've completely rewritten the page and added a lot of new content. The old combinatorial definitions are still hidden in comments, waiting for someone less tired than I am right now to put them back in in a reasonable way. - Gauge 05:37, 30 Apr 2005 (UTC)


The statement that a simplicial set is a contravariant functor from the opposite of the simplicial category Δop is wrong. Either one shoud say that it is a (covariant) functor from Δop, or a contravariant functor from Δ (no opposite).

It also seems misleading to say that simplicial sets are used in algebraic situations where CW complexes would typically not apply, since the realization of a simplicial set is ALWAYS a CW complex- by its very definition.

I suggest adding information about the uses of simplicial sets. Something like the following:

" Simplicial sets were originally used to give precise and convenient descriptions of classifying spaces of groups. This idea was vastly extended by Grothendiecks idea of considering classifying spaces of categories, and in particular by Quillens work of algebraic K-theory. In this work, which earned him a Field's medal, Quillen developed surprisingly efficient methods for manipulating infinite simplicial sets. Later these methods has been used in other areas on the border between algebraic geometry and topology. For instance, the André-Quillen homology of a ring is a "non-abelian homology", defined and studied in this way.

Both the algebraic K-theory and the André-Quillen homology are defined using algebraic data to write down a simplicial set, and then taking the homotopy groups of this simplicial set. Sometimes one simply defines the algebraic K-theory as the space.

Simplicial methods are often useful when you want to prove that a space is a loop space. The basic idea is that if G is a group, BG it's classifying space then G is homotopy equivalent to the loop space ΩBG. If BG itself is a group, we can iterate the procedure, and G is homotopy equivalent to the double loop space Ω2B(BG). In case G is an Abelian group, we can actually iterate this infinitely many times, and obtain that G is an infinite loop space.

Even if X is not an Abelian group, it can happen that it has a composition which is sufficiently commutative so that one can use the above idea to prove that X is an infinite loop space. In this way, one can prove that the algebraic K-theory of a ring, considered as a topological space, is an infinite loop space.

References:

Daniel Quillen: Higher algebraic K-theory: I. In: H. Bass (ed.): Higher K-Theories. Lecture Notes in Mathematics, vol. 341. Springer-Verlag, Berlin 1973. ISBN 3-540-06434-6

G.B. Segal, Categories and cohomology theories, Topology, 13, (1974), 293 - 312. "

Marcel Bökstedt

Hi Marcel. Thank you for your comments. The loop space stuff was particularly interesting to me as I am still learning the ropes. Regarding: It also seems misleading to say that simplicial sets are used in algebraic situations where CW complexes would typically not apply, since the realization of a simplicial set is ALWAYS a CW complex- by its very definition— the point I was trying to make is that it may be difficult to treat certain geometric objects like the etale site on a scheme directly as CW-complexes, whereas it is always possible to take a simplicial nerve (which can then be realized as a CW-complex, if desired). There is probably a better way to word this, and I'd be glad to hear any insights you have to share about it. With your permission, I think I might try to work your text above into some kind of history/motivation section. Best wishes, Gauge 00:36, 11 August 2006 (UTC)