# Abstract Relation between Presehaves and Simplicial Sets

Every presheaf (let's say on a topological space) comes with restriction maps. The open sets of a topological space are ordered by inclusion and these inclusions yield the restrictions. Now a sheaf satisfies a gluing condition: that you can glue along elements which coincide on common restrictions.

Every simplicial object (let's say a simplicial set) comes with face maps. The simplex category is ordered by faces & degeneracies and these maps yield simplicial maps. Now a Kan complex satisfies a gluing* condition: that you can glue along simplices which coincide on common faces.

Is there a deeper theoretical framework to relate these 2 notions? I guess that this is the case, and that it is rather trivial.

Side-question: Can we define "degeneracies" for presheaves?

Ideas?

*it's not a gluing condition, but "somehow similar" (see answers below)

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Well, a simplicial object in C is a C-valued presheaf on the simplicial category. Is this what you mean? – Qiaochu Yuan Nov 12 '09 at 20:47
No, that is obvious: both are contravariant functors. – Konrad Voelkel Nov 12 '09 at 23:13
But since I'm learning about model categories, the answers below are what I expected, and this is really helpful for me to understand this. It's hard to choose one as "answer"! – Konrad Voelkel Nov 12 '09 at 23:14

I don't really get how you see the Kan Horn filling condition as a gluing condition.

But sheaves and Kan simplicial sets play parallel roles in their categories if you look at it through model category theory: In both situations you have an endofunctor replacing a presheaf by a sheaf, a simplicial set by a Kan set respectively. Both categories have a model structure - that is a bunch of data allowing to handle the formal inversion of morphisms which are called weak equivalences.Both times you have a morphism from the old to the new object which is a weak equivalence, this process is called fibrant replacement and is formalized in the theory of model categories.

In the presheaf case the weak equivalences are those morphisms which become isomorphisms after applying the sheafification functor. If you formally invert these, the resulting category is equivalent to the category of sheaves.

In the simplicial set case the weak equivalences are those maps which induce isomorphisms of homotopy groups after applying geometric realization. If you formally invert those you get a category equivalent to the homotopy category of spaces.

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Do you mean "Quillen equivalent" when you say "equivalent"? – Konrad Voelkel Nov 12 '09 at 23:17
No, I mean actually equivalent. "Quillen equivalent" is a notion for model categories, i.e. categories having all this extra structure but in which you have not yet inverted the "weak equivalences". If you then invert these you get the "homotopy category" of the model category. Quillen equivalence of two model categories implies equivalenc of their homotopy categories - in general it is not itself an equivalence of categories! The statements in the last two paragraphs relate the homotopy categories to other, equivalent, categories - no model structures are involved there. – Peter Arndt Nov 13 '09 at 1:10
@peter: I think I see what the OP means by saying that a kan complex is like a glueing condition. Suppose that you have two n-simplices glued about a face. Then we may map the two simplices glued about a face into some horn (a horn with some faces removed). The composite of the two simplicies glued into the horn, then into the n+1 simplex is a cofibration. Thus we may lift it to a map from he n-simplex. Just a Guess. – Spice the Bird Nov 4 '11 at 17:58

Yes, so:

A simplicial set is indeed precisely a presheaf on the simplex category.

There are various model category structures on categories of presheaves in general and on simplicial sets in particular.

With respect to the standard model structure on simplicial sets the Kan complexes are precisely the fibrant-and-cofibrant objects.

With respect to the local model structure on presheaves on a site the sheaves are precisely the fibrant-and-cofibrant objects.

There is a very useful combination of these two statements:

A simplicial presheaf is a presheaf on the product category of the simplex category and some site.

in the local projective model structure on simplicial presheaves the fibrant objects are precisely those simplicial presheaves that are Kan-complexes over each object of the site and that satisfy the oo-version of the sheaf condition ("descent"): these are the (hypercomplete) oo-stacks on the given site.

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Thank you! I didn't understand before, that sheaves are precisely fibrant-cofibrant in the local model structure of presehaves on a site. Now the picture is much clearer. – Konrad Voelkel Nov 12 '09 at 23:22
Does the local projective model structure yield the same homotopy category as the local model structure? – Konrad Voelkel Nov 12 '09 at 23:23
If by "the local model structure" you mean the local injective one: yes. See the big diagram at ncatlab.org/nlab/show/model+structure+on+simplicial+presheaves which lists all the Quillen equivalences between the diferrent model structures. All Quillen equivalent structures yield the same homotopy category, in particular. But it's even better: they even yield equivalent (oo,1)-categories. – Urs Schreiber Nov 12 '09 at 23:31

The Kan condition isn't exactly like the sheaf condition: the Kan condition allows you to "glue" (as you put it) in certain cases, but the result is not unique.

A better analogy to the Kan condition in sheaf theory might be the notion of a flasque sheaf: a sheaf F is flasque if for all subsets V of U, all sections of F over V extend to sections over U.

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It should be mentioned that if you require that the Kan condition have unique gluing, you get exactly those Kan complexes that are nerves of groupoids. – Eric Wofsey Nov 12 '09 at 22:55