TITLE: Simplicial objects
QUESTION [38 upvotes]: 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?

REPLY [8 votes]: 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.