Friday, February 05th, 2010 | Author: Konrad Voelkel
We look at the model structure Voevodsky and Morel use in their 1999 IHES paper and discuss 1.2, 1.3, 1.4, 1.5, 1.6, 1.8, 1.9, 1.10. There is nothing difficult or particularly interesting, but you might want to look up some specific issue or reference.
I wrote another posting that explains what an enriched model category, enriched over a monoidal model category is; we turn to simplicial model categories in this post. There, I also explain the notion of monoidal and enriched model categories beside some notions of simplicial sets and topoi, the most important being for now:
a simplicial model category is just an enriched model category which is enriched over the monoidal model category of simplicial sets.
but details are also to be found below.
The simplicial model structure on simplicial sheaves on a topos
In Definition 1.2, for every small site , a model structure on is defined:
- The weak equivalences are the stalkwise (pointwise) weak equivalences
- The cofibrations are the monomorphisms
- The fibrations are defined via the right lifting property with respect to acyclic cofibrations
Remark 1.3 is a technical subtlety. If you happen to have a conservative set of points of a topos , then weak equivalence of a morphism of sheaves on can be tested pointwise: , where denotes the weak equivalences in the standard model structure of simplicial sets. A conservative set of points is just a set of points that is a conservative family of functors, which is by definition, that the product functor is a conservative functor.
A functor is conservative if it reflects isomorphisms. That means, isomorphism implies isomorphism for each morphism .
This technical lemma is used later in the text, but the homotopy sheaves are not, so I guess you can forget the proof details when reading the text for the first time.
See also: conservative functor in nLab
Theorem 1.4 (the structure defined by is a model category structure) cites the result of Corollary 2.7 in
Jardine: Simplicial Presheaves, in no. 47 J.Pure Applied Math, 1987 which is originally due to Joyal. Since the article is behind a paywall, I'll give you a rough idea:
- (MC1), (MC2) and (MC3) are deduced from the model structure on simplicial sets.
- (MC4) relies on the fact that the morphism from a presheaf to its associated sheaf is a weak equivalence and then applying the axiom for with the global fibration and topological weak equivalence model structure. (MC4) for is proved with a trick that uses (MC5).
- (MC5) is essentially a small object argument.
The corresponding homotopy category of on is written .
See also: small object argument in nLab
Proper model categories
Remark 1.5 states that the model structure is a proper one. The proof is available in
Jardine, J.F.: Stable homotopy theory of simplicial presheaves, in no. 39 Can. Math. J, 1987 which is available for free here.
A simplicial model category is proper if
- (P1) the pullback of a weak equivalence along a fibration is always a weak equivalence,
- (P2) the pushout of a weak equivalence along a cofibration is always a weak equivalence.
(P1) is proved for simplicial sets via fibrant replacement, such that one has a cartesian diagram up to weak equivalence, and then application of K. Brown's coglueing lemma, which is Lemma 1 on page 428 of
Brown, K.: Abstract Homotopy Theory and Generalized Sheaf Cohomology, in Vol. 186 Transactions of the American Mathematical Society, 1973 which you can download from the nLab for free.
(P2) is proved for simplicial sets in a dual fashion, using the fact that simplicial sets are always cofibrant and a dual of Brown's coglueing lemma.
For simplicial presheaves on a topos, the proofs are similar. For (P1), fibrant replacement yields a cartesian diagram (up to weak equivalence) in which all objects are locally fibrant simplicial presheaves (which form a category of fibrant objects) and the coglueing argument can be applied. For simplicial sheaves, (P1) and (P2) follow since the associated sheaf morphism is a weak equivalence.
It should be mentioned that (P1) is also called right proper and similarly (P1) left proper.
See also: proper model category in nLab
Functorial fibrant replacements (1.6)
(MC5) demands in particular, that every morphism is functorially factorizable into a fibration after an acylic cofibration.
A resolution on a site (which carries a model structure) is defined to be a functor and a transformation such that for every simplicial sheaf , the object is fibrant and is an acyclic cofibration.
Indeed, if is a morphism, we can factorize it into an acyclic cofibration followed by a fibration. Rename the acyclic cofibration and the object , then is a fibration, thus fibrant. Voilà - since (MC5) demands this to be functorial, the functor/transformation conditions for a resolution are fulfilled.
It should be clear that this works the same way for cofibrant replacements, although we won't need this here, since in the simplicial model structure we're looking at on , all objects are cofibrant.
See also: Kan fibrant replacement in nLab
Simplicial model categories
For every two objects , we defined
is a simplicial set because is a cosimplicial object. If you take an object as constant simplicial sheaf in degree 0, you can look at , which is just the simplicial set of sections for the simplicial sheaf . Now we have to see that this enrichment is compatible with the model structure. This is done in Remark 1.9. resp. Lemma 1.8. The proof indication for Lemma 1.8. is to prove 1) via points of . This is easy if you already know that the standard model structure on simplicial sets is a simplicial model structure (the model category of simplicial sets enriched over the monoidal model category of simplicial sets), which is not too hard to prove.
If you already know about the "subtleties" in the definition of simplicial model categories (maybe from my article about simplicial model categories), skip the next two paragraphs.
A category is a simplicial model category if it is a model category that is enriched over simplicial sets, that satisfies the additional axioms (Quillen):
- (SM0): for all and all finite simplicial sets , and exist.
- (SM7): If is a cofibration and a fibration, then
is a fibration of simplicial sets, which is trivial if either or is trivial. (The S denotes the simplicial mapping object of ).
(SM0) is also phrased " is powered and copowered" and sometimes already included in the definition of an enriched model category (like I did in my article about simplicial model categories). (SM7) is also phrased "the copower functor is a left Quillen bifunctor" and sometimes already included in the definition of an enriched model category (like I did, again). So, if you take the "modern" definition of a model category enriched over a monoidal model category, those axioms are already included (I put them in here just because they will show up in the literature and also because you might not have read my article about the definition of simplicial model categories).
Lemma 1.10, different notions of equivalence are the same
For fibrant and a morphism, these three statements are equivalent:
- is a simplicial homotopy equivalence,
- is a weak equivalence,
- is a weak equivalence.
The proof indication is mostly a list of references, so let's have a more detailed look, which will then finish this posting.
factorise the weak equivalence into a cofibration followed by an acyclic fibration . Then is a weak equivalence again (by 2-out-of-3). By an argument in Quillen's Homotopical Algebra (Corollary 2.5), obtain a retraction of by the lift in the diagram
and then get a simplicial homotopy from to by the lift in the diagram
and now is a simplicial homotopy inverse of . To actually obtain a simplicial homotopy inverse of , we're going to build a simplicial homotopy inverse of . For this, observe that all objects are cofibrant (since cofibrations are by definition just monomorphisms), and that the dual statement to what we just proved is that a trivial fibration between cofibrant objects is a simplicial homotopy equivalence.
What is ? What is ? you might ask. The object is just the simplicial set , whose geometric realisation in looks like the interval , hence the name (and I used this notation here because it's the same as in Quillen's book). The object is the internal mapping object . If this remains unclear, you might want to read some introduction to enriched category theory.
We will not try to construct a weak homotopy equivalence but a homotopy equivalence:
Using the definition of for and , you'll see the canonical isomorphism . Now take a simplicial homotopy inverse to the map and choose a simplicial homotopy between and . This yields a map which, composed with the canonical isomorphism above, is the homotopy between and we're looking for. The other composition is handled the same way.
From SGA4 6.8.2 we learn that every point of has an associated functor , where is the category of neighbourhoods (French: voisinages) of . A neighbourhood is a couple where and . The cofiltrant category of neighbourhoods of admits a small cofinite full subcategory, so by abstract nonsense the functor is a pro-object in . A pro-object is, by definition, just a functor from a small cofiltered category to (think of it as a diagram to form a projective limit, hence the name). Let's write the pro-object , hiding the small cofinal full subcategory of in the indices.
Now for a point , is a filtering colimit (=projective limit) of all , thus a filtering colimit of weak equivalences. We conclude that is itself a weak equivalence. Since this holds for every point, is a weak equivalence.