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Sometimes people say "If you don't like the word 'topos', just think the category of Sets", but I'm not sure to what extent this analogy holds.

The real question here is, do simplicial object in a topos have the structure of a model category? I'm just not sure if you really need the geometric realization functor (and topological spaces) to define the usual model category structure on simplicial sets.

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Yes, there is a general theory of model structures on simplicial sheaves (due to Joyal), and a theory of simplicial presheaves as well (Jardine). Jardine has a paper "Simplicial presheaves" that develops this in the latter case. The cofibrations are the monomorphisms of simplicial presheaves. The weak equivalences are the "local weak equivalences" (i.e. inducing equivalences on the sheaves of homotopy groups associated). –  Akhil Mathew Nov 8 '11 at 22:48
    
Note that topological spaces are nowhere necessary to define weak equivalences of simplicial sets. A map $X \to Y$ of simplicial sets is a weak equivalence if and only if for every Kan complex $K$, the map $\underline{\hom}(Y, K) \to \underline{\hom}(X, K)$ (internal homs) is a homotopy equivalence. One can obtain the existence of the model structure purely combinatorially. I believe a generalized theory of this is done in Cisinski's thesis, but I have not read it. Anyway, you may find helpful the nLab page: ncatlab.org/nlab/show/model+structure+on+simplicial+presheaves –  Akhil Mathew Nov 8 '11 at 22:49
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@Akhil - "model structures on simplicial sheaves" this is not quite the same as saying 'model structures on simplicial objects in a topos', because a topos might not be a Grothendieck topos. In particular, one could take the category of finite sets (a non-Grothendieck elementary topos) and consider simplicial objects in it. –  David Roberts Nov 8 '11 at 23:07
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I'm not sure I understand the question. Here is an example: let sfSets be the category of simplicial objects in finite sets; this is some sort of topos. You can define geometric realization of such things just fine, and use it to call a map a weak equivalence iff its realization is a weak equivalences of spaces. This is not a model category, even in Quillen's original sense: any object which has non-finite homotopy groups will not have a finite replacement in sfSets. –  Charles Rezk Nov 8 '11 at 23:17
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Every finite- complete and cocomplete category is a model category, where the weak equivalences are just the isomorphisms. So, yes. –  Charles Rezk Nov 9 '11 at 0:03

2 Answers 2

up vote 8 down vote accepted

As has been established in the comments the answer for a general topos is no, while for a Grothendieck topos it is yes, by the work of Joyal.

The general question of when one can transfer a model structure on a category based on Sets to an arbitrary Grothendieck topos is beautifully adressed in Tibor Beke's articles on "Sheafifiable Homotopy Model Categories", available here. The examples include simplicial objects, cyclic objects and groupoid and category objects.

The proofs really use the assumption that you are in a Grothendieck topos (e.g. he chooses a site defining the given topos and uses the existence of morphisms to the topos of sets, plus the existence of the necessary colimits to interpret infinitary geometric logic, possibly even accessibility somewhere) and I don't think his arguments can be saved for more general toposes. Anyway the papers are worth a look and contain several interesting remarks about the role of accessibility in establishing model structures, giving a good perspective on your original question.

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Beke also makes a remark on establishing fibration category structures by the same techniques... –  Peter Arndt Nov 9 '11 at 1:02
    
How about for a topos bounded over another topos which does admit the relevant model structure on the diagram category of interest? –  David Roberts Nov 9 '11 at 1:30
    
Sounds reasonable, but I don't have the time to really check it... –  Peter Arndt Nov 9 '11 at 7:55

As pointed out, the answer is no – because we cannot hope to have fibrant replacements in general. On the other hand, by following the programme of van Osdol [1977, Simplicial homotopy in an exact category], it is possible to quickly establish a weaker result:

Theorem. For any regular category $\mathcal{S}$ (e.g. an elementary topos), the category of internal Kan complexes in $\mathcal{S}$ is a category of fibrant objects where the fibrations are the internal Kan fibrations and the trivial fibrations are the internal trivial Kan fibrations. (By Ken Brown's lemma, this suffices to determine the weak equivalences.)

Here, "internal" refers to the internal logic of regular categories: for example, an internal trivial Kan fibration in $\mathcal{S}$ is a morphism between simplicial objects in $\mathcal{S}$ such that the matching morphisms (à la Reedy) are regular epimorphisms. Note however that we are using the "external" notion of simplicial objects.

In the special case where $\mathcal{S}$ is a sheaf topos, the internal Kan fibrations and internal trivial Kan fibrations turn out to be the same thing as Jardine's local fibrations and local trivial fibrations. (See Theorem 1.12 in [1987, Simplicial presheaves].) It follows that the weak equivalences in the sense above are the same as Jardine's local weak equivalences.

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