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Are there general principles when a simplicial object is a (co)limit of other simplicial objects level by level, but is not a (co)limit when considering the entire simplicial structure?

Objects can be groups or algebras etc, and limits here can something as simple as products, coproducts or limits, colimits.

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Theo Johnson-Freyd asked a similar question: mathoverflow.net/questions/17953/… – Tom Leinster Apr 3 '10 at 11:48

Limits and colimits can be computed point-wise in functor categories, such as the category of simplicial objects of any category. That is, if you have a functor category $\mathcal{C}^I$ ($I = \Delta$ in your case) and a functor $F: P \longrightarrow \mathcal{C}^I$, in particular, for every object $p$ in $P$ you have a functor $F_p : I \longrightarrow \mathcal{C}$. Then $\varinjlim_p F_p$ is an object of $\mathcal{C}^I$; that is, a functor $I \longrightarrow \mathcal{C}$, whose value on objects $i$ in $I$ is

$$ (\varinjlim_p F_p)(i) = \varinjlim_p (F_p(i)) \ . $$

This is true as far as the colimit on the right exists for every object $i$ in $I$. An analogous statement applies for limits in functor categories.

For limits, you can find the result in MacLane, "Categories for the working mathematician", chapter V, section 3, theorem 1 (see at the end of the proof also). For colimits, see "Sheaves in geometry and logic: a first introduction to topos theory", by Saunders Mac Lane,Ieke Moerdijk, p.40.

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I think, that the levelwise colimit agrees with the colimit in the category Func$(\Delta,\mathcal{C}$. As the colimit may be viewed as a functor from the category of directed systems over $\mathcal{C}$ to $\mathcal{C}$ the differentials/degeneracies in a directed system of objects in Func($\Delta,\mathcal{C})$ induces differentials/degeneracies on the levelwise colimit.

So the levelwise colimit is really a simplicial object. And then one can show, that it satisfies the universal property of the colimit in the category Func$(\Delta,\mathcal{C})$.

The same proof also works for limits, or any other construction (I can imagine) defined by an universal property, that doesn't involve the simplicial structure.

I think, there is also a more conceptual proof of this, but I can't remember.

However it is not true, that limits commute with geometric realization: Any set may be viewed as a constant functor $\Delta\rightarrow \mathcal{Sets}$. A levelwise inverse limit of constant functors is again a constant functor and its realization is discrete, while the inverse limit of the realizations need not be discrete.

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Just to add a point of detail: colimits in Func(Delta, C) are calculated pointwise just as long as C has the relevant colimits. It can happen that "weird", non-pointwise, colimits exist in Func(Delta, C) if C fails to have enough colimits. See my answer to mathoverflow.net/questions/17953 – Tom Leinster Apr 4 '10 at 2:46

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