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model structure on functors

Context

Model category theory

model category

Definitions

  • category with weak equivalences

  • weak factorization system

  • homotopy

  • small object argument

  • resolution

  • Morphisms

    • Quillen adjunction

    • Universal constructions

      • homotopy Kan extension

      • homotopy limit/homotopy colimit

      • Bousfield-Kan map

      • Refinements

        • monoidal model category

        • enriched model category

        • simplicial model category

        • cofibrantly generated model category

        • algebraic model category

        • compactly generated model category

        • proper model category

        • cartesian closed model category, locally cartesian closed model category

        • stable model category

        • Producing new model structures

          • on functor categories (global)

          • on overcategories

          • Bousfield localization

          • transferred model structure

          • Grothendieck construction for model categories

          • Presentation of (,1)(\infty,1)-categories

            • (∞,1)-category

            • simplicial localization

            • (∞,1)-categorical hom-space

            • presentable (∞,1)-category

            • Model structures

              • Cisinski model structure
              • for \infty-groupoids

                for ∞-groupoids

                • on topological spaces

                  • Strom model structure?
                • Thomason model structure

                • model structure on presheaves over a test category

                • on simplicial sets, on semi-simplicial sets

                • model structure on simplicial groupoids

                • on cubical sets

                • on strict ∞-groupoids, on groupoids

                • on chain complexes/model structure on cosimplicial abelian groups

                  related by the Dold-Kan correspondence

                • model structure on cosimplicial simplicial sets

                • for nn-groupoids

                  • for n-groupoids/for n-types

                  • for 1-groupoids

                  • for \infty-groups

                    • model structure on simplicial groups

                    • model structure on reduced simplicial sets

                    • for \infty-algebras

                      general

                      • on monoids

                      • on simplicial T-algebras, on homotopy T-algebras

                      • on algebas over a monad

                      • on algebras over an operad,

                        on modules over an algebra over an operad

                      • specific

                        • model structure on differential-graded commutative algebras

                        • model structure on differential graded-commutative superalgebras

                        • on dg-algebras over an operad

                        • model structure on dg-modules

                        • for stable/spectrum objects

                          • model structure on spectra

                          • model structure on ring spectra

                          • model structure on presheaves of spectra

                          • for (,1)(\infty,1)-categories

                            • on categories with weak equivalences

                            • Joyal model for quasi-categories

                            • on sSet-categories

                            • for complete Segal spaces

                            • for Cartesian fibrations

                            • for stable (,1)(\infty,1)-categories

                              • on dg-categories
                              • for (,1)(\infty,1)-operads

                                • on operads, for Segal operads

                                  on algebras over an operad,

                                  on modules over an algebra over an operad

                                • on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations

                                • for (n,r)(n,r)-categories

                                  • for (n,r)-categories as ∞-spaces

                                  • for weak ∞-categories as weak complicial sets

                                  • on cellular sets

                                  • on higher categories in general

                                  • on strict ∞-categories

                                  • for (,1)(\infty,1)-sheaves / \infty-stacks

                                    • on homotopical presheaves

                                    • model structure for (2,1)-sheaves/for stacks

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                                      Contents

                                      Idea

                                      The model category structures on functor categories are models for (∞,1)-categories of (∞,1)-functors.

                                      For CC a model category and DD any small category there are two “obvious” ways to put a model category structure on the functor category [D,C][D,C], called the projective and the injective model structures. For completely general CC, neither one need exist. The projective model structure exists as long as CC is cofibrantly generated, while the injective model structure exists as long as CC is combinatorial.

                                      A related kind of model structure is the Reedy model structure/generalized Reedy model structure on functor categories, which applies for any model category CC, but requires DD to be a very special sort of category, namely a Reedy category/generalized Reedy category.

                                      In the special case that C=C = sSet is the classical model structure on simplicial sets the projective and injective model structure on the functor categories [D,SSet][D,SSet] are described in more detail at global model structure on simplicial presheaves.

                                      Definition

                                      Definition

                                      For CC a combinatorial model category or, in the projective case, just a cofibrantly generated model category, and DD a small category there exist the following two (combinatorial) model category structures on the functor category [D,C][D,C]:

                                      • the projective structure [D,C] proj[D,C]_{proj}: weak equivalences and fibrations are the natural transformations that are objectwise such morphisms in CC.

                                      • the injective structure [D,C] inj[D,C]_{inj}: weak equivalences and cofibrations are the natural transformations that are objectwise such morphisms in CC.

                                      More generally, if CC is in addition a simplicial model category and DD a smooth sSet-enriched category, then the sSet-enriched functor category, also denoted [D,C][D,C], carries the above two model strutures.

                                      Properties

                                      In all of the following, let S\mathbf{S} be an excellent model category. The standard example is the model structure on simplicial sets, sSet QuillensSet_{Quillen}.

                                      Let DD (and D 1D_1, D 2D_2, …) be a combinatorial S\mathbf{S}-enriched model category.

                                      Moreover, CC in the following is assumed to be either an ordinary small category, or, more generally, it is a small S\mathbf{S}-enriched category.

                                      If S=\mathbf{S} = sSet Quillen{}_{Quillen} and CC is an ordinary small category, then then model structures discussed here are instances of the model structure on simplicial presheaves. If CC is itself sSetsSet-enriched, then they are instances of the model structure on sSet-enriched presheaves.

                                      General

                                      Proposition

                                      The projective and injective structures [C,D] proj[C,D]_{proj} and [C,D] inj[C,D]_{inj}, def.

                                      The existence of the unenriched model structure apears as HTT, prop. A.2.8.2 The enriched case is HTT, prop. A.3.3.2 and the remarks following that. The statement about properness appears as HTT, remark A.2.8.4.

                                      Cofibrant generation

                                      Proposition

                                      Let CC be an ordinary small category.

                                      The cofibrations in [C,A] proj[C, A]_{proj} are generated from (i.e. are the weakly saturated class of morphisms defined by) the morphisms of the form

                                      Id C(c,)i:C(c,)aC(c,)b Id_{C(c,-)}\cdot i : C(c,-)\cdot a \to C(c,-) \cdot b

                                      for all cCc \in C and i:abi : a \to b a generating cofibration in AA. Here the dot denotes the tensoring of AA over sets, i.e. C(c,)aC(c,-)\cdot a is the functor that sends cCc' \in C to the coproduct C(c,c)A\coprod_{C(c,c')} A of |C(c,c)||C(c,c')| copies of AA.

                                      In particular, every cofibration if [C,A] proj[C,A]_{proj} is in particular a cofibration in [C,A] inj[C,A]_{inj}. Similarly, every fibration in [C,A] inj[C,A]_{inj} is in particular a fibration in [C,A] proj[C,A]_{proj}

                                      This is argued in the beginning of the proof of HTT, lemma A.2.8.3.

                                      For TopTop-enriched functors, this is (Piacenza 91, section 5). For details see at classical model structure on topological spaces – Model structure on enriched functors.

                                      Relation to other model structures

                                      Corollary

                                      The identity functors

                                      [D,C] injIdId[D,C] proj [D,C]_{inj} \stackrel{\overset{Id}{\leftarrow}}{\underset{Id}{\to}} [D,C]_{proj}

                                      form a Quillen equivalence (with Id:[D,C] proj[D,C] injId : [D,C]_{proj} \to [D,C]_{inj} being the left Quillen functor).

                                      If DD is a Reedy category this factors through the Reedy model structure

                                      [D,C] injIdId[D,C] ReedyIdId[D,C] proj [D,C]_{inj} \stackrel{\overset{Id}{\leftarrow}}{\underset{Id}{\to}} [D,C]_{Reedy} \stackrel{\overset{Id}{\leftarrow}}{\underset{Id}{\to}} [D,C]_{proj}

                                      Functoriality in domain and codomain

                                      Proposition

                                      The functor model structures depend Quillen-functorially on their codomain, in that for

                                      D 1RLD 2 D_1 \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D_2

                                      a S\mathbf{S}-enriched Quillen adjunction between combinatorial S\mathbf{S}-enriched model categories, postcomposition induces S\mathbf{S}-enriched Quillen adjunctions

                                      [C,D 1] proj[C,R][C,L][C,D 2] proj [C,D_1]_{proj} \stackrel{\overset{[C,L]}{\leftarrow}}{\underset{[C,R]}{\to}} [C,D_2]_{proj}

                                      and

                                      [C,D 1] inj[C,R][C,L][C,D 2] inj. [C,D_1]_{inj} \stackrel{\overset{[C,L]}{\leftarrow}}{\underset{[C,R]}{\to}} [C,D_2]_{inj} \,.

                                      Moreover, if (LR)(L \dashv R) is a Quillen equivalence, then so is ([C,L][C,R])([C,L] \dashv [C,R]).

                                      For the case that CC is a small category this is (Lurie, remark A.2.8.6), for the enriched case this is (Lurie, prop. A.3.3.6).

                                      The Quillen-functoriality on the domain is more asymmetric.

                                      Proposition

                                      For p:C 1C 2p : C_1 \to C_2 a functor between small categories or an S\mathbf{S}-enriched functor between S\mathbf{S}-enriched categories, let

                                      (p !p *p *):[C 2,D]p *p *p ![C 1,D] (p_! \dashv p^* \dashv p_*) : [C_2,D] \stackrel{\overset{p_!}{\leftarrow}}{\stackrel{\overset{p^*}{\to}}{\underset{p_*}{\leftarrow}}} [C_1,D]

                                      be the adjoint triple where p *p^* is precomposition with pp and where p !p_! and p *p_* are left and right Kan extension along pp, respectively.

                                      Then we have Quillen adjunctions

                                      (p !p *):[C 1,D] projp *p ![C 2,D] proj (p_! \dashv p^*) : [C_1,D]_{proj} \stackrel{\overset{p_!}{\to}}{\underset{p^*}{\leftarrow}} [C_2,D]_{proj}

                                      and

                                      (p *p *):[C 1,D] injp *p *[C 2,D] inj. (p^* \dashv p_*) : [C_1,D]_{inj} \stackrel{\overset{p^*}{\leftarrow}}{\underset{p_*}{\to}} [C_2,D]_{inj} \,.

                                      For CC not enriched this appears as (Lurie, prop. A.2.8.7), for the enriched case it appears as (Lurie, prop. A.3.3.7).

                                      Remark

                                      In the sSetsSet-enriched case, if p:D 1D 2p : D_1 \to D_2 is an weak equivalence in the model structure on sSet-categories, then these two Quillen adjunctions are both Quillen equivalences.

                                      Proposition

                                      For CC a combinatorial simplicial model category, the (∞,1)-category presented by [D,C] proj[D,C]_{proj} and [D,C] inj[D,C]_{inj} under the above assumptions is the (∞,1)-category of (∞,1)-functors Func(D,C )Func(D,C^\circ) from the ordinary category DD to the (,1)(\infty,1)-category presented by CC.

                                      See (∞,1)-category of (∞,1)-functors for details.

                                      Relation to homotopy Kan extensions/limits/colimits

                                      Often functors DCD \to C are thought of as diagrams in the model category CC, and one is interested in obtaining their homotopy limit or homotopy colimit or, generally, for p:DDp : D \to D' any functor, their left and right homotopy Kan extension.

                                      These are the left and right derived functors HoLan:=𝕃p 1HoLan := \mathbb{L} p_1 and HoRan:=p *HoRan := \mathbb{R} p_* of

                                      [D,C] projp ![D,C] proj [D,C]_{proj} \stackrel{p_!}{\to} [D',C]_{proj}

                                      and

                                      [D,C] injp *[D,C] inj [D,C]_{inj} \stackrel{p_*}{\to} [D',C]_{inj}

                                      respectively.

                                      For more on this see homotopy Kan extension. For the case that D=*D' = * this reduces to homotopy limit and homotopy colimit.

                                      Examples

                                      Examples of cofibrant objects in the projective model structure are discussed at

                                      References

                                      The projective model structure on Top QuillenTop_{Quillen}-enriched functors is discussed in

                                      • Robert Piacenza section 5 of Homotopy theory of diagrams and CW-complexes over a category, Can. J. Math. Vol 43 (4), 1991 (pdf)

                                        also chapter VI of Peter May et al., Equivariant homotopy and cohomology theory, 1996 (pdf)

                                      See also

                                      • Alex Heller, Homotopy in functor categories, Transactions of the AMS, vol 272, Number 1, July 1982 (JSTOR)

                                      General review and discussion includes

                                      The projective model structure for functors to sSet on a large domain is discussed in

                                      See also

                                      Last revised on July 24, 2017 at 14:30:04. See the history of this page for a list of all contributions to it.