Holmstrom Model category examples

Lots of examples related to brave new rings are mentioned in here. Other, more standard examples:

A nonexample: (from Vezzosi lectures in Seville): there is no reasonable model structure on $cdga(k)$ if k is a field of char p.

Functor categories

One might ask in general about model structures on functor cats $C^B$ , where $C$ is a MC. If $B$ is so-called direct, can define WEs and fibrations objectwise, and get a model structure. If $B$ is so-called inverse, can define WEs and cofibs objectwise, and get a model structure. See Hovey 5.1 for details and properties. If $B$ is a so-called Reedy category, can define a model structure by objectwise WEs, and a more complicated def of fibs and cofibs. See Hovey 5.2. Note that $\Delta$ and its opposite are Reedy cats.

Consider the category of functors from a category $C$ to $Sset$ . We can define the projective model structure on this functor category as follows: WEs and fibrations are defined pointwise, while a cofibration is a “retract of a cellular inclusion” (see Dundas, p.40).

See Weibel’s obituary for Thomason - were these ideas ever written up?

http://nlab.mathforge.org/nlab/show/global+model+structure+on+functors

Simplicial sets

(Ref: Hovey). We define the set $I$ as the set of inclusions $\partial \Delta [n] \to \Delta [n]$ for $n \geq 0$ . Define $J$ to be the set of inclusions $\Lambda^r[n] \to \Delta[n]$ for $n>0, \ 0 \leq r \leq n$ . A map $f$ is a cofibration iff it is in $I-cof$ . A map is a (Kan) fibration iff it is in $J-inj$ . A map $f$ is a weak equivalence iff its geometric realization is a weak equivalence of topological spaces. The maps in $J-cof$ are called anodyne extensions.

From the definition, it follows that: A map is a cofibration iff it is injective. Hence every simplicial set is cofibrant. Also, every cofibration is a relative I-cell complex.

Thm: The category $Sset$ is a finitely generated model category with generating cofibrations I, generating trivial cofibrations J, and the above WEs. Same for pointed simplicial sets (fibs, cofibs, WEs are those in $Sset$ ).

Simplicial groups, and simplicial groupoids

See Garzon, Miranda, Osorio

Topological spaces

A map $f: X \to Y$ in $Top$ is a weak equivalence if

\pi_n(f, x): \pi_n(X,x) \to \pi_n(Y, f(x) )

is an isomorphism for all $n \geq 0$ and all $x \in X$ . Let $I'$ be the set of boundary inclusions $S^{n-1} \to D^n$ for all $n \geq 0$ and let $J$ be the set of inclusions $D^n \to D^n \times [0,1]$ , $x \mapsto (x,0)$ . We define a cofibration to be an element of $I'-cof$ and a (Serre) fibration to be an element of $J-inj$ .

Of course, every homotopy equivalence is a weak equivalence.

Theorem (Hovey p. 57): There is a finitely generated model structure on $Top$ with $I'$ as the set of generating cofibrations, J as the set of generating trivial cofibrations, and the WEs as above. Every object of $Top$ is fibrant, and the cofibrations are retracts of relative cell complexes.

There is a very similar statement for $Top_*$ . There are also model structures on the category of k-spaces and on the category of compactly generated spaces, as well as on the pointed version of these cats. See Hovey pp. 58 for details.

See the last paragraph of this nLab entry and also this entry for other model structures on Top, e.g. the mixed model structure.

Chain complexes of comodules over a Hopf algebra

See Hovey, section 2.5.

Chain complexes of R-modules

We define a model structure on $Ch(R)$ as follows. For any $R$ -module $M$ we define $S^n(M)$ to be the complex with the module $M$ placed in degree $n$ . We also define $D^n(M)$ to be the complex with $M$ in degree $n$ and $n-1$ , and the identity as differential between them. Dropping $M$ in this notation means that $M=R$ .

Let $I$ be the set of evident injections $S^{n-1} \to D^n$ , and let $J$ be the maps $0 \to D^n$ . Define a map to be a fibration if it is in $J-inj$ and a cofibration if it is in $I-cof$ . As usual, a map is a WE if it induces an isomorphism on homology.

From the above definition, it follows that a map is fibration if and only if it is surjective in each degree. A map is a trivial fibration iff it is in $I-inj$ . If $A$ is a cofibrant chain complex, then $A_n$ is projective for all $n$ , and a bounded below complex of projectives is cofibrant. In general, a map is a cofibration iff it is a dimensionwise split inclusion with cofibrant cokernel.

Chain complexes of R-mods, with the injective model structure

Here is another model structure on $Ch(R)$ , called the injective model structure. We define a map to be an injective fibration if it has the RLP wrt all maps that are both injections and WEs.

Theorem: The injections, injective fibrations, and WEs are part of a cofibrantly generated model structure on $Ch(R)$ . The injective fibrations are the surjections with fibrant kernel. Every fibrant object is a complex of injectives. Every bounded above complex of injectives is fibrant. The injective trivial fibrations are the surjections with injective kernel; a complex is injective iff it is fibrant and acyclic.

Modules over a quasi-Frobenius ring

A ring $R$ is called quasi-Frobenius if the projective and injective $R$ -modules coincide. Examples include the group ring of a finite group over a field. For such a ring, there is a cofibrantly generated model structure on $R-mod$ where the cofibrations are the injections, the fibrations are the surjections, and the WEs are the “stable equivalences”.

Other examples

J.F. Jardine, “A closed model structure for differential graded algebras”, Cyclic Cohomology and Noncommutative Geometry, Fields Institute Communications, Vol. 17, AMS (1997), 55-58.

http://www.ams.org/mathscinet-getitem?mr=764018 Golasinski: Model structures on $pro-Cat$ and $pro_0-Cat$

Larusson on equivalence relations

nLab entries on model structure: Model cat, and various entries beginning with model structure on

For model structure on DG-algebras, see Gelfand and Manin: Methods of homological algebra, Chapter 5

Simplicial groupoids, see Dwyer: Homotopy th and simplicial groupoids. See also http://www.ncatlab.org/nlab/show/model+structure+on+presheaves+of+simplicial+groupoids

Pointed simplicial spaces, see Dwyer-Kan-Stover paper, as well as its review for a comparison with Reedy structure.

Isaksen has a few papers on limits and model structures on pro-cats. In particular, for any proper model category C he produces a model structure on pro-C. I don’t have these papers electronically, but they probably exist.

Harper: Homotopy theory of modules over operads etc. ArXiv. This paper studies the existence of model category structures on algebras and modules over operads in monoidal model categories.

Larusson: The cat of complex manifolds can be embedded into a model cat in a way such that a manifold is cofibrant iff it is Stein, and fibrant iff it is Oka. See AMS Notice Jan 2010: What is an Oka manifold. Original article not referenced.

Ostvaer stuff on Cstar-algebras

nLab page on Model category examples

Created on June 9, 2014 at 21:16:13 by Andreas Holmström