Talk:Simplicial set

contravaraint
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)

still backwards
The statement that a simplicial set is a contravariant functor from the opposite of the simplicial category $$\Delta^{op}$$ is wrong. Either one shoud say that it is a (covariant) functor from $$\Delta^{op}$$, or a contravariant functor from $$\Delta$$ (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 $$\Omega BG$$. If $$BG$$ itself is a group, we can iterate the procedure, and $$G$$ is homotopy equivalent to the double loop space $$\Omega^2 B(BG)$$. 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

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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&mdash; 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)


 * Marcel, I have added your paragraph to the article. Thanks! AxelBoldt (talk) 23:21, 20 September 2014 (UTC)

complex
It is formerly called semi-simplicial complex, or full semi-simplicial complex. I got confused when I redirected from semi-simplicial set. Yunzhi (talk) 00:44, 3 October 2009 (UTC)

Face and degeneracy maps
It appears to me that the formulas defining the face and degeneracy maps are not quite correct. Check Weibel's book. — Preceding unsigned comment added by 128.211.178.28 (talk) 21:33, 21 November 2013 (UTC)


 * I independently came to the same conclusion. As written down, they use the strings which where above defined to be the objects (and hence the domains of the function) as arguments. So one should not use the notation with arrows, to denote the collection of numbers which are the elements of that objects. 129.247.247.240 (talk) 15:07, 12 December 2013 (UTC)


 * The notation is fixed now. AxelBoldt (talk) 20:43, 8 December 2021 (UTC)

In the definition, it is claimed that the morphisms of the simplex category are the order-preserving functions. Do they mean increasing or strictly increasing? If strictly increasing, then there are no degeneracy maps. If just increasing, then there is more than only the maps composed of face and degeneracy maps, e.g. the image of $$[n] \rightarrow [n+1]: x \mapsto 0$$. If we only want maps constructed from face and degeneracy maps, we have to equip Delta with all injective and all surjective order-preserving maps.


 * They mean increasing functions, and all of these can indeed be gotten by composing face and degeneracy maps. The map you are describing can be gotten by first going from [n] all the way down to [0] with a sequence of &sigma;0 maps, and then from [0] to [n+1] with a sequence of &delta;1 maps. AxelBoldt (talk) 05:01, 19 September 2014 (UTC)

Undefined notation in "Face and degeneracy maps" section
The definitions of the face and degeneracy maps use notation that, to me, seems not to be widely known (at least, one wouldn't expect to be taught this notation in an undergraduate math career), yet is not explained. Eg.
 * di (0 &rarr; &hellip; &rarr; n) = (0 &rarr; &hellip; &rarr; i &minus; 1 &rarr; i + 1 &rarr; &hellip; &rarr; n).

By some luck, I was able to decipher what is meant (I think). (Background: I am a professional functional analyst). However, I think that an explanation of this notation, or at least a reference to such an explanation, should be included. — Preceding unsigned comment added by 86.184.84.173 (talk) 18:50, 1 May 2014 (UTC)


 * ✅ AxelBoldt (talk) 20:45, 8 December 2021 (UTC)

Clearer definition
The arrow notation $$0 \rightarrow 1 \rightarrow \ldots \rightarrow n$$ is bad, especially without explanation, because it could be read as a curried function with multiple parameters. I suggest phrasing it: "Where Δ is the simplex category, whose objects are finite (non-empty?) sections of the natural numbers, i.e. sets of the form [n] = {0, 1, ..., n}".


 * I got rid of all the arrows. AxelBoldt (talk) 05:01, 19 September 2014 (UTC)

Too abstract
Everything is too abstract and complicated here. A simplicial set is just a finite set with a collection of subsets that is closed by inclusion. 2001:B07:A5B:3825:F874:1D10:77A9:45C7 (talk) 15:54, 17 October 2023 (UTC)