Simplicial objects

Question

How should one think about simplicial objects in a category versus actual objects in that category? For example, both for intuition and for practical purposes, what's the difference between a [commutative] ring and a simplicial [commutative] ring?

Answer 1

One could say many things about this, and I hope you get many replies! Here are some remarks, although much of this might already be familiar or obvious to you.

In some vague sense, the study of simplicial objects is "homotopical mathematics", while the study of objects is "ordinary mathematics". Here by "homotopical mathematics", I mean the philosophy that among other things say that whenever you have a set in ordinary mathematics, you should instead consider a space, with the property that taking pi_0 of this space recovers the original set. In particular, this should be done for Hom sets, so we should have Hom spaces instead. This is formalized in various frameworks, such as infinity-categories, simplicial model categories, and A-infinity categories. Here "space" can mean many different things, in these examples: infinity-category, simplicial set, or chain complex respectively.

For intuition, it helps to think of a simplicial object as an object with a topology. For example, a simplicial set is like a topological space, a simplicial ring is like a topological ring etc. The precise statements usually takes the form of a Quillen equivalence of model categories between the simplicial objects and a suitable category of topological objects. Simplicial sets are Quillen equivalent to compactly generated topological spaces, and I think a similar statement holds if you replace sets by rings, although I am not sure if you need any hypotheses here.

If you like homological algebra, it helps to think of a simplicial object as analogous to a chain complex. The precise statements are given by various generalizations of the Dold-Kan correspondence. For simplicial rings, they should correspond to chain complexes with a product, more precisely DGAs. Again, one has to be a bit careful with the precise statements. I think the following is true: Simplicial commutative unital k-algebras are Quillen equivalent to connective commutative differential graded k-algebras, provided k is a Q-algebra.

A remark about the word "simplicial": A simplicial object in a category C is a functor from the Delta category into C, but for almost all purposes the Delta category could be replaced with any test category in the sense of Grothendieck, see this nLab post for some discussion which doesn't use the terminology of test categories.

Since you used the tag "derived stuff" I guess you are already aware of Toen's derived stacks. Some of his articles have introductions which explain why one would like to use simplicial rings instead of rings. See in particular his really nice lecture notes from a course in Barcelona last year.

I tried to write a blog post on some of this a while ago, there might be something useful there, especially relating to motivation from algebraic geometry.

Answer 2

Just to amplify the very good points made by Andreas Holmstrom above:

a) it just so happens that simplices are one of the best, at least best understood, models for higher homotopical structure and most every simplicial object that you will run into is secretly a model for an oo-groupoid with extra structure.

b) In particular this is true for the case of simplicial abelian groups, and monoids of these: simplicial abelian rings. The underlying simplicial set of a simplicial group is a necessarily a Kan complex, hence an oo-groupoid. So simplicial groups are oo-groupoids with strict group structure on them. In particular simplicial abelian groups are therefore fully abelian oo-groupoids: special kinds of connective spectra.

c) See the introduction at nLab:Dold-Kan correspondence: this correspondence allows to not only understand all these simplicial objects as models for oo-groupoids with extra structure and property, but also lots of objects in homological algebra as being just yet another (computationally convenient) repackaging of that information.

And when you feel at home with a,b,c), open Lurie's "Stable (oo,1)-Cateories" and see his (oo,1)-version of the Dold-Kan correspondence there to see the full truth...

Answer 3

In case you are an algebraic geometer and are you used to thinking about a commutative ring in terms of its spectrum it might be helpful to imagine the spectrum of a simplicial commutative ring A as the spectrum of \pi_0(A) together with a fuzzy cloud of generalized nilpotents. This can actually be made precise: For a simplicial commutative ring A there is a closed immersion Spec(\pi_0 (A)) -> Spec (A), and their underlying point sets are the same. So the relationship between spectra of simplicial rings to spectra of commutative rings is really much the same as the relationship between general schemes and reduced schemes.

Answer 4

This is an answer to `What can you do with simplicial objects' instead of how to think of them.

The main context in which I think about simplicial objects is Cohomological Descent. This is, as Martin Olsson put it, Cech cohomology on steroids. It is an extremely useful tool (for instance it makes it totally easy to prove things like GAGA for Deligne-Mumford stacks). Brian Conrad has an excellent set of notes here. SGA IV, expose 5 is another (much more abstract) reference. Deligne's Hodge III is another source. There he uses coh. descent to put Hodge structures on singular (complex analytic) varieties.

Answer 5

It depends on the context. Simplicial objects are often used to "resolve" objects in nonabelian contexts to get a good notion of derived functors. For example, in Andre-Quillen cohomology you take a ring (viewed as a discrete simplicial set) and resolve it by finding a weakly equivalent simplicial ring that's levelwise free. Then you can apply various functors to this (such as abelianization) and get good notions of derived functors.

A simplicial object {X_n} in particular always includes two maps from X₁ to X₀ and you can think of the simplicial object as representing a lift of the coequalizer of these two maps to something living in a derived context.

Simplicial objects are often equivalent in some fashion to topological objects; e.g. simplicial commutative rings are equivalent (in a model-category sense) to topological commutative rings.

Sometimes there's a "realization" functor from simplicial objects to regular objects, but in general simplicial objects play their own role.

Answer 6

This is a good place to mention the notion of a "Grothendieck test category". This is a small A category which has the property that presheaves of sets on A, with an appropriate class of weak equivalences, models the homotopy theory of spaces. So the simplicial indexing category Δ is a test category, but there are others, such as the category which indexes cubical sets. I would guess that whenever one needs to use simplicial rings (say), you could replace simplicial with any test category (though I don't know that anyone has worked this sort of thing out).

These notions are developed in some papers of Cisinski, and a good introduction (which gives the definition of test category) is the paper of Jardine. Of course, this doesn't really answer your question on how to think about simplicial objects, but perhaps it puts it in a broader context.

Answer 7

The definition of a simplicial object is a functor $X_\bullet\colon \Delta^{op}\to A$ where $\Delta$ is the simplicial category and $A$ is your favorite category. So the easiest answer is that a simplicial object in a category is a sequence of objects in that category together with morphisms $d_i$ and $s_j$ that satisfy a bunch of relations.

We can also think of a simplicial object $X_\bullet$ as an element of $Fun(\Delta^{op}, A)$, the category of functors from $\Delta^{op}$ to $A$. In this context, it is easy to define a morphism between simplicial objects. It maps $X_n\to Y_n$ and commutes with the $d_i$'s and $s_j$'s.

Shameless planar algebra plug: Simplicial objects are also really cool in tandem with an adjoint functor pair. You can use this machinery to get a pictorial representation of the simplicial category using Temperley-Lieb (string) diagrams. In fact, planar algebras are great examples of simplicial vector spaces, although there's a lot more structure too...