It seems that a simplicial set should be thought as a space/thing, with encoded building information. The set of $n$-simplicies is the set of n-dimensional pieces and the boundary and degeneracy maps tell you how they fit together.

With this viewpoint though, I find myself puzzled about how to think of simplicial objects in general. Provided this is still a valid way of looking at things - now, our 'collections of $n$-dim info' themselves form objects in a category. I can think of two possible ways of interpreting this:

$\bullet$ our building/construction now has pieces with more structure for structure's sake. (Algebraic operations, for example, give 'moving parts' and interactions.)

$\bullet$ Or, $K \in \mathcal{C}^{\Delta^{op}}$ can be somehow regarded as a construction 'within' $\mathcal{C}$. (With the $n$-simplices forming a sort of 'moduli object' for all $n$.)

So, are these good ways of looking at simplicial objects? If so, may you please legitimise them?. If not, may you please disabuse me of them (and maybe suggest an alternative)? I'm looking for ways to intrinsically interpret simplicial objects (of rings, schemes, etc.), without reinterpreting them in a new category (like, say, via Dold-Kan type results).

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    $\begingroup$ I have no idea how would anyone legitimize anything here, but the second bullet point is precisely how I think people view simplicial objects. «Hey, I know how these things work in Top so I'll just pretend everything is Top» Surprisingly, i works qquite well. $\endgroup$ – Mariano Suárez-Álvarez Feb 20 '15 at 3:24
  • $\begingroup$ Yeah, it seems harmless to think of a simplicial abelian group as something like a topological abelian group: for instance in the latter case the singular simplicial set lifts through abelian groups. $\endgroup$ – Kevin Arlin Feb 20 '15 at 5:23

In my view, passing to simplicial objects is all about introducing higher-dimensional structure, which gives us "maneuvering room" to do some useful things. For instance, not every $k$-algebra is a free $k$-algebra (i.e. a polynomial algebra over $k$), but every $k$-algebra is weakly homotopy equivalent to a simplicial $k$-algebra that is degreewise free. The construction is essentially a non-additive version of the construction of free resolutions in homological algebra: one way of thinking about the higher-dimensional data is to regard the $(n+1)$-simplices as specifying "relations" between the $n$-simplices.

There is a precise sense in which we should think of simplicial objects as being instructions for gluing things together. Let $\mathcal{C}$ be a locally small category with colimits for countable diagrams. The geometric realisation of a simplicial object $X_{\bullet}$ in $\mathcal{C}$ with respect to a functor $T : \mathbf{\Delta} \times \mathcal{C} \to \mathcal{C}$ is the object $\left| X \right|$ in $\mathcal{C}$ equipped with a bijection $$\mathcal{C} (\left| X \right|, Y) \cong [\mathbf{\Delta}^\mathrm{op} \times \mathbf{\Delta}, \mathbf{Set}] (\mathbf{\Delta} (-, \bullet), \mathcal{C} (T (-, X_{\bullet}), Y))$$ that is natural in $Y$, where $\mathbf{\Delta} (-, \bullet) : \mathbf{\Delta}^\mathrm{op} \times \mathbf{\Delta}, \mathbf{Set}$ is the usual hom functor and $T (-, X_{\bullet}) : \mathbf{\Delta} \times \mathbf{\Delta}^\mathrm{op} \to \mathcal{C}$ is the functor $(n, m) \mapsto T (n, X_m)$. This turns out to amount to saying that $\left| X \right|$ fits into a certain coequaliser diagram in $\mathcal{C}$ of the form below, $$\coprod_{\phi : [n] \to [m]} T (n, X_m) \rightrightarrows \coprod_n T (n, X_n) \to \left| X \right|$$ where the upper arrow is defined on components by $T (\phi, X_m) : T (n, X_m) \to T (m, X_m)$ and the lower arrow is defined on components by $T (n, \phi^*) : T (n, X_m) \to T (n, X_n)$.

The choice of $T : \mathbf{\Delta} \times \mathcal{C} \to \mathcal{C}$ depends on the context. For example, for $\mathcal{C} = \mathbf{Top}$, we take $T (n, Z) = \Delta^n \times Z$, where $\Delta^n$ is the standard $n$-simplex. If $\mathcal{C}$ is the category of simplicial objects in a locally small category $\mathcal{A}$ with countable colimits, then it is standard to take $T (n, Z)_m = \mathbf{\Delta} (m, n) \odot Z_m$ (i.e. $\mathbf{\Delta} (m, n)$-many copies of $Z_m$). In this situation, simplicial objects in $\mathcal{C}$ are bisimplicial objects in $\mathcal{A}$, and it turns out that for any bisimplicial object $X_{\bullet, \bullet}$ in $\mathcal{A}$, the geometric realisation is given by $\left| X \right|_n = X_{n, n}$. Thus, if $X_{\bullet, \bullet}$ is a simplicial diagram of "good" simplicial objects in $\mathcal{A}$, then $\left| X \right|_{\bullet}$ is also a "good" simplicial object in $\mathcal{A}$. That is one reason why we don't have to pass to bisimplicial objects to resolve simplicial objects.

If that's still too abstract, it might be helpful to focus on the case of simplicial objects in algebra. Let $\mathcal{A}$ be a category with a "forgetful" functor $U : \mathcal{A} \to \mathbf{Set}$ and let $\mathbf{s} \mathcal{A}$ be the category of simplicial objects in $\mathcal{A}$. Of course, we get a "forgetful" functor $U : \mathbf{s} \mathcal{A} \to \mathbf{sSet}$, so we can declare a morphism in $\mathbf{s} \mathcal{A}$ be a weak equivalence if its image in $\mathbf{sSet}$ is a weak homotopy equivalence. If $U : \mathcal{A} \to \mathbf{Set}$ has a left adjoint $F : \mathbf{Set} \to \mathcal{A}$, then $U : \mathbf{s} \mathcal{A} \to \mathbf{sSet}$ also has a left adjoint $F : \mathbf{sSet} \to \mathbf{s} \mathcal{A}$. We think of objects in the image of $F : \mathbf{Set} \to \mathcal{A}$ as being "free" objects. At this point, it is easy to show that every object $A$ in $\mathcal{A}$ can be resolved by a degreewise "free" simplicial object, i.e. there is a weak equivalence $X_{\bullet} \to A$ in $\mathbf{s} \mathcal{A}$ where $X_{\bullet}$ is a simplicial object such that each $X_n$ is a "free" object. Moreover, in favourable contexts (e.g. when $U : \mathcal{A} \to \mathbf{Set}$ factors through the forgetful functor $\mathbf{Grp} \to \mathbf{Set}$, or slightly more generally, when $U : \mathbf{s} \mathcal{A} \to \mathbf{sSet}$ factors through the full subcategory of Kan complexes), it can be shown that $F : \mathbf{sSet} \to \mathbf{s} \mathcal{A}$ preserves weak equivalences, and it then follows that every simplicial object in $\mathcal{A}$ can be resolved by a degreewise "free" simplicial object.


Assume that your notion of "object" is internalizable into arbitrary topoi. This is almost always the case. Then a simplicial object of the given type is usually (always?) an object of the given type internal to the topos of simplicial sets. For example, a simplicial group is a group internal to the topos of simplicial sets. A simplicial module (over a simplicial ring) is a module internal to the topos of simplicial sets (over an internal ring). Before you struggle with technical problems of simplicial groups, try to think about groups internal to arbitrary categories, and some (not all, of course!) difficulties will disappear.

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    $\begingroup$ That procedure only works well for models of cartesian theories. As is well known, a sheaf of local rings (= internal local ring in the topos of sheaves) is not literally a presheaf of local rings that happens to be a sheaf. $\endgroup$ – Zhen Lin Apr 8 '15 at 5:45
  • $\begingroup$ But this is caused by the sheafification process, right? I only talk about presheaves (indexed by $\Delta$). $\endgroup$ – Martin Brandenburg Apr 8 '15 at 10:25
  • $\begingroup$ Yes, things work much better for presheaf toposes. $\endgroup$ – Zhen Lin Apr 8 '15 at 10:42

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