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Math 549. Model categories. Fall, 2001.
Lecture notes.

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September 6

Discussed limits and colimits.  We will alsoways be working in a category Cthat has arbitrary limits and colimits.  Then I gave the definition (1.1.3) of a  model structure on such a category. In it there are three special classes of morphisms called weak equivalences, fibrations and cofibrations, and they satisfy 4 axioms.

After repeating the 4 axioms for a model category and briefly discussing pushouts and pullbacks, I gave the construction for the factorization axiom (the trickiest of the four) in the category Top of topological spaces.  Then I started a lengthy discussion of how one associates a homotopy category to a model category.  This included the definition of cylinder and path objects.

Went over the proof of Prop 1.2.5(iii), which says that left homotopy is an equivalence relation when the domain is cofibrant, and 1.2.5(v) which says under the same hypotheses left homotopy implies right homotopy and that any path object can be used for the latter  These imply Corollary 1.2.7, which says that in the subcategory consisting of all objects which are both fibrant and cofibrant, left and right homotopy are equivalent, both are equivalence relations, and any cylinder and path objects can be used for them.

These require a lemma that is not stated in the book saying that a map from a cofibrant object to a cylinder for it is always a trivial cofibration.

I decided to skip sections 1.3 and 1.4; I may come back to them later.  Chapter 2 treatss various examples of model catgories including the classical case Top.   Before one can study even the simplest interesting example (namely the stable category of modules over a Frobenius ring) efficiently one needs some preliminaries.  I talked about small and finite objects (Definitions 2.1.3 and 2.1.4) relative to a collection of maps in a category. Then I gave the definition of a relative I-cell complex (2.1.9), which generalizes the notion of a relative CW-complex.

Finally I gave the definition of a cofibrantly generated model category (2.1.17). In it the class of (trivial) fibrations is determined by relatively little data.  Most model categories one encounter in real life have this property.  Theorem 2.1.19 (to be discussed in our next meeting, on October 1) characterizes the data that are needed.
 

I began a discussion of a model structure on the category of chain complexes of R-modules for a ring R, following [DS95].  In it weak equivalences are maps inducing isomorphisms in homology, cofibrations are monomorphisms with projective cokernels, and  fibrations are maps that are surjective in positive degrees.  Showing that the required liftings exist involves some tedious diagram chasing.

The construction of functorial factorizations is more interesting because it illustrates the small object argument.  A preliminary step is showing that a map is a fibration iff it has the right lifting property with respect to the inclusion 0 --> D^n for each n>0, and a trivial fibration iff it has the right lifting property with respect to the inclusion S^{n-1} --> D^n for each n>=0. We denote these two sets of maps by J and I respectively.

An object in a category is said to finite if it behaves well in terms of maps to direct limits.  In the case of sets this coincides with the usual notion of finiteness, and in the case of modules finite means finitely presented.  The sets of maps J and I above have finite domains.  The small object argument (Prop 7.17 of [DS95] and Thm 2.1.14 of Hovey) says that given such a set of maps I, any map can be naturally factored as a relative I-cell complex (i.e., a cofibration built up from pushouts involving maps in I) followed by a trival fibration. This is proved by constructing the intermediate object explicitly.  In the case of the category chain complexes, this construction for the sets J and I above gives the two factorizations needed for a model structure.

Prop 7.3 of [DS95] identifies  homotopy classes of maps between certain objects in this category with Ext groups.  What can be said about homotopy classes of maps between more general chain complexes?
 

Discussed the category Ch(B) of chain complexes of comodules over a commutuative Hopf algebra B (which could be graded) over a field k.    The relevant example of this in algebraic topology is the mod p homology of a space or spectrum, which is always a comodule over the dual Steenrod algebra A_*.  A comodule over B is automatically a module over the dual algebra B^* (eg the mod p Steenrod algebra A).  This module is always tame, ie each element generates a finite dimensional submodule.  The A-module used in the definition of the root invariant is not tame.

Discussed the model structure on the category Ch(B). Naively one would define this in the same way as for the category of chain complexes of B^*-modules, but there are several reasons why this does not work.  For one thing the notion of homology equivalence is not subtle enough, so weak equivalences are defined more finely in 2.5.10. Another problem is that the complexes D^n(B^*) and S^n(B^*) are not in the category because B^* is not in tame as a module over itself unless it is finite dimensional. Hence the maps in I and J (see the notes for 10/2 above) have to be replaced by certain collections of inclusions of finite dimensional chain complexes.

Discussed the classical model structure on Top, the category of topological spaces.  This is treated by Hovey in Chapter 2.4 and by Dywer-Spalinski in Section 8.  Here the term weak equivalence has its usual homotopy theoretic meaning, fibrations are Serre fibrations, and cofibrations are maps having the left lifting property with respect to every Serre fibration which is also a weak equivalence.  The resulting homotopy category is that of CW-complexes.

Discussed simplicial sets, using Dwyer's 1998 Barcelona lectures (#96 on his home page) as a reference for motivating the definition.  An abstract simplicial complex K is a set of vertices with certain subsets designated as simplices.  It has a geometric realization (ie a topological space functorially associated with it) |K|. Unfortunately geometric realization respects neither products nor pushouts. The problem with products can be solved by using ordered simplicial complexes, in which the vertex set is partially ordered in such a way that each simplex is well ordered.  The product in OSC, the category of such complexes, is preserved by geometric realization, but pushouts are not.  (The proof that products are preserved can also be found in  May and Eilenberg-Steenrod.) In order to preserve pushouts we replace  ordered simplicial complexes by simplicial sets, which are set valued functors on a certain category related to standard simplices.

Proved that geometric realization on Sp, the category of simplicial sets, preserves colimits (and in particular pushouts) by observing that the geometric realization functor is the left adjoint of the singular complex functor S from Top to Sp, and showing that any left adjoint preserves colimits.  Also defined the simplicial set BC for a small category C; its geometric realization |BC| is the classifying space of C.  Defined simplicial objects in a general category C and the chain complex associated with a simplicial abelian group.  The chain complex associated with the free abelian group on the simplical set SX for a topological space X is the singular chain complex of X.

Gave an example due to Dwyer showing that geometric realization need not commute with infinite products.  It involved a certain product of 1-dimensional simplicial sets.  [There is a simpler example involving discrete sets.  A discrete space A is the geometric realization of the A-valued constant simplical set, i.e., the functor which sends each object of the simplex category to A. The geometric realization of an infinite product of such sets is the infinite product endowed with the discrete topology, while the product of the realizations is the same set with the product topology.] Then I discussed the proof that that geometric realization does commute with finite products, which is 3.1.8 in Hovey.  Finally I began discussing the  model structure on the category of simplicial sets.

Spent most of the lecture reviewing the construction of a cofibrantly generated model category from data about its weak equivalences, generated cofibrations and generating trivial cofibrations.  This is stated as Theorem 2.1.19 in Hovey and summarized in the handout "Cofibrantly generated model categories " available on the home page.  Then I proved 3.2.2, which says that a map of simplicial sets is 1-1 iff it is a cofibration.

Proved 3.2.3, which says that every anodyne extension is a trivial cofibration. This is condition (iv) of Theorem 2.1.19 applied to simplicial sets.  This was followed by Lemma 3.2.4, which says that geometric realization preserves finite limits.  This requires a lemma from category theory (not stated by Hovey but proved by Mac Lane as Theorem V.2.1) reducing the problem from finite limits to equalizers and finite products.  Our goal now is to establish conditions (v) and (vi) of 2.1.19, i.e, that every I-injective map is a trivial fibration.  The first step is 3.2.5, which says that the geometric realization of such a map is always a fibration of topological spaces.

Proved 3.2.5, which says that every I-injective map is a trivial fibration.  Before we can prove the converse we need to define the homotopy groups of a fibrant simplicial set or Kan complex.  The set of connected components \pi_0(X) is defined in Lemma 3.4.2.  For the higher groups we want to mimic one version of the topological definition, i.e.,

\pi_n(X, x) := \pi_0 (Map((D^n, S^{n-1}), (X, x))),

where Map((D^n, S^{n-1}), (X, x)) is the fiber of the restriction map

Map(D^n, X) --> Map(S^{n-1}, X)

over the constant map sending S^{n-1} to the point x.

To do this we need some knowledge of mapping complexes. The simplicial set Map(K, L) is defined on page 77 by

Map(K, L)_n := Hom_{SSet}(K \times \Delta[n], L).

This imples that

Map(\Delta[0], L)_n = Hom_{SSet}(\Delta[n], L) = L_n

so Map(\Delta[0], L) = L.  We need to know that the map

Map(\Delta[n], X) --> Map(\partial\Delta[n], X)

is a fibration for fibrant K.  This is a special case of Theorem 3.3.1.

I stated Theorems 3.3.1 and 3.3.2 and explained the fact that they are equivalent. This requires the exponential law, which says that

Hom_{SSet}(A, Map(K, L)) = Hom_{SSet}(A \times K, L)).

Our goal now is to show that every trivial fibration of simplicial sets is I-injective, which is the converse of 3.2.5, discussed on 10/30.  We say simplicial set is acyclic if it is nonemepty and all of its homotopy groups are trivial.  Homotopy groups (actually homotopy sets) were defined in 3.4.4 using Theorem 3.3.1, which guarantees that a certain map is a fibration.  Prop. 3.4.7 says that a map from an acyclic set to a point is I-injective.  Cor 3.5.2 extends this to locally trivial fibrations X --> Y (3.5.1), i.e., fibrations which restrict to a product fibration over each simplex of Y.  [This is a point where the tradition use of the term 'trivial fibration', i.e., a Cartesian product, clashes with Hovey's use of the same term for a fibration which is a weak equivalence.]

Lemma 3.5.3 (which I did not prove) says that homotopic maps induce fiber homotopy equivalent fibrations.  In particular (3.5.4) any fibration over the standard n-simplex is fiber homotopically trivial.    A minimal fibration X --> Y (3.5.5) is one in which any two simplices in X having the same boundary and the same image in Y are identical.  [This is incorrect; the simplices are identical if they have the same boundary and they are fiber homotopic to each other;. This latter condiution is straonger than having the same image in Y.]  Cor 3.5.7 says that a minimal fibration is locally trivial, Theorem 3.5.9 says that any fibration is the composite of an I-injective retraction followed by a minimal fibration.  These imply 3.5.10, which says that any fibration with acyclic fibers is I-injective.
 

Proved 3.5.3, which says the homotopic maps induce fiber homotopically equivalent fibrations.  Then I proved 3.5.6 and used it to show that a fiber homotopy equivalence between minimal fibrations must be an isomorphism.

Proved Lemma 3.5.8 (each nondegenerate simplex is determined by its boundary) and sketched the proof of part of  Theorem 3.5.9, which says that every fibration has a minimal retract.  These are stated and proved as 10.2 and 10.3 in  [GJ97] (Simplicial Homotopy Theory by P.G. Goerss and J.F.Jardine). Their proof of the latter is more explicit than Hovey's.  This result is the last major obstacle to showing that SSet is a model category.

Discussed Definitions 4.1.1-3, all of which have to do with monoidal categories, i.e., categories endowed with a tensor product.  (These definitions have been simplified somewhat as explained by Hovey in his errata.)  This lead to a brief discussion of 2-categories, which are defined in 1.4.1 and will be discussed more next time.

Gave the definition of a 2-category  (1.4.1).  What I thought was a typo in Hovey was actually a typo in my notes.  Discussed symmetric monoidal categories.

Defined an adjunction of two variables (4.1.12) and a Quillen adjunction of same (4.2.1) and related notions leading up to the definition of a monoidal model category (4.2.6).

Talked about left and right derived functors and Quillen equivalences, as discussed in \S1.3 of the book.

The object of Chapter 5 is to show that the homotopy category for any model category is a module over the one for the category of simplicial sets.  I explained how to put a model structure on the category of functors from a Reedy category (of which the simplicial category and its opposite are both examples) to a model category.

Talked about Theorems 5.4.8 and 5.5.3, which say that the homotopy category of any model category is a module over the one for simplicail sets, with all the nice module structure one could hope for.

This page was last revised on December 11, 2001, by Doug Ravenel .