nLab
Bousfield-Friedlander model structure

Contents

Context

Stable Homotopy theory

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

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

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

for (,1)(\infty,1)-operads

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

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

Contents

Idea

The Bousfield-Friedlander model structure (Bousfield-Friedlander 78, section 2) is a model structure for spectra, specifically it is a standard model structure on sequential spectra in simplicial sets. An immediate variant works for sequential spectra in topological spaces, see at model structure on topological sequential spectra.

As such, the Bousfield-Friedlander model structure presents the stable (infinity,1)-category of spectra of stable homotopy theory, hence, in particular, its homotopy category is the classical stable homotopy category.

Background on sequential spectra

Sequential pre-spectra

Write S 1Δ[1]/Δ[1]S^1 \coloneqq \Delta[1]/\partial\Delta[1] for the minimal simplicial circle. Write

:sSet */×sSet */sSet */ \wedge \;\colon\; sSet^{\ast/} \times sSet^{\ast/} \longrightarrow sSet^{\ast/}

for the smash product of pointed simplicial sets.

Definition

A sequential prespectrum in simplicial sets, or just sequential spectrum for short (or even just spectrum), is

  • an \mathbb{N}-graded pointed simplicial set X X_\bullet

  • equipped with morphisms σ n:S 1X nX n+1\sigma_n \colon S^1 \wedge X_n \to X_{n+1} for all nn \in \mathbb{N}.

A homomorphism f:XYf \colon X \to Y of spectra is a sequence f :X Y f_\bullet \colon X_\bullet \to Y_\bullet of homomorphisms of pointed simplicial sets, such that all diagrams of the form

S 1X n S 1f n S 1Y n σ n X σ n Y X n+1 f n+1 Y n+1 \array{ S^1 \wedge X_n &\stackrel{S^1 \wedge f_n}{\longrightarrow}& S^1 \wedge Y_n \\ \downarrow^{\mathrlap{\sigma_n^X}} && \downarrow^{\mathrlap{\sigma_n^Y}} \\ X_{n+1} &\stackrel{f_{n+1}}{\longrightarrow}& Y_{n+1} }

commute.

Write SeqSpec(sSet)SeqSpec(sSet) for this category of sequential spectra.

Example

For XSeqSpec(sSet)X \in SeqSpec(sSet) and KK \in sSet, hence K +sSet */K_+ \in sSet^{\ast/} then XK +X \wedge K_+ is the sequential spectrum degreewise given by the smash product of pointed objects

(XK +) n(X nK +) (X \wedge K_+)_n \coloneqq (X_n \wedge K_+)

and with structure maps given by

S 1(X nK +)(S 1X n)K +σ nK +X n+1K +. S^1 \wedge (X_n \wedge K_+) \simeq (S^1 \wedge X_n) \wedge K_+ \stackrel{\sigma_n \wedge K_+}{\longrightarrow} X_{n+1}\wedge K_+ \,.
Proposition

The category SeqSpecSeqSpec of def. becomes a simplicially enriched category (in fact an sSet */sSet^{\ast/}-enriched category) with hom objects [X,Y]sSet[X,Y]\in sSet given by

[X,Y] nHom SeqSpec(sSet)(XΔ[n] +,Y). [X,Y]_n \coloneqq Hom_{SeqSpec(sSet)}(X\wedge \Delta[n]_+,Y) \,.
Definition

The stable homotopy groups of a sequential spectrum XX, def. , is the \mathbb{Z}-graded abelian groups given by the colimit of homotopy groups of geometric realizations of the component spaces

π (X)lim kπ +k(|X n|). \pi_\bullet(X) \coloneqq \underset{\longrightarrow}{\lim}_k \pi_{\bullet+k}({\vert X_n \vert}) \,.

This constitutes a functor

π :SeqSpec(sSet)Ab . \pi_\bullet \;\colon\; SeqSpec(sSet) \longrightarrow Ab^{\mathbb{Z}} \,.
Definition

A morphism f:XYf \colon X \longrightarrow Y of sequential spectra, def. , is called a stable weak homotopy equivalence, if its image under the stable homotopy group-functor of def. is an isomorphism

π (f):π (X)π (Y). \pi_\bullet(f) \;\colon\; \pi_\bullet(X) \longrightarrow \pi_\bullet(Y) \,.

Omega-spectra

Definition

A Omega-spectrum is a sequential spectrum XX, def. , such that after geometric realization/Kan fibrant replacement the (smash product dahsv\dahsv pointed mapping space)-adjuncts

|X n||X n+1| |S 1| {\vert X_n\vert} \stackrel{}{\longrightarrow} {\vert X^{n+1}\vert}^{{\vert S^1\vert}}

of the structure maps |σ n|{\vert \sigma_n\vert} are weak homotopy equivalences.

Remark

If a sequential spectrum XX is an Omega-spectrum, def. , then its colimiting stable homotopy groups, def. , are attained as the actual homotopy groups of its components:

π k(X){π k|X 0| ifk0 π 0|X k| ifk<0. \pi_k(X) \simeq \simeq \left\{ \array{ \pi_k {\vert X_0 \vert} & if\; k \geq 0 \\ \pi_0 {\vert X_k \vert} & if \; k \lt 0 } \right. \,.
Definition

The canonical Ω\Omega-spectrification QXQ X of a sequential spectrum XX of simplicial sets, def. , is the operation of forming degreewise the colimit of higher loop space objects Ω()() S 1\Omega(-)\coloneqq (-)^{S^1}

(QX) nlim kSingΩ k|X n+k|, (Q X)_n \coloneqq \underset{\longrightarrow}{\lim}_{k } Sing \Omega^k {\vert X_{n+k}\vert } \,,

where SingSing denotes the singular simplicial complex functor.

This constitutes an endofunctor

Q:SeqSpec(sSet)SeqSpec(sSet). Q \;\colon\; SeqSpec(sSet) \longrightarrow SeqSpec(sSet) \,.

Write

η:idQ \eta \;\colon\; id \longrightarrow Q

for the natural transformation given in degree nn by the (||Sing)({\vert-\vert}\dashv Sing)-adjunction unit followed the 0-th component map of the colimiting cocone:

(η X) n:X nSing|X n|ι 0lim kSingΩ k|X n+k|. (\eta_X)_n \;\colon\; X_n \longrightarrow Sing{\vert X_n\vert} \stackrel{\iota_0}{\longrightarrow} \underset{\longrightarrow}{\lim}_{k } Sing \Omega^k {\vert X_{n+k}\vert } \,.
Proposition

The spectrification of def. satisfies

  1. QXQ X is an Omega-spectrum, def. ;

  2. η X:XQX\eta_X \colon X \longrightarrow Q X is a stable weak homotopy equivalence, def. ;

  3. if for a homomorphims of sequential spectra f:XYf \colon X \longrightarrow Y each f nf_n is a weak homotopy equivalence, then also each (QX) n(Q X)_n is a weak homotopy equivalence;

  4. (Qη X)(Q\eta_X) is degreewise a weak homotopy equivalence.

Corollary

A homomorphism of sequential spectra, def. , is a stable weak homotopy equivalence, def. , precisely if its spectrification QfQ f , def. , is degreewise a weak homotopy equivalence.

The strict model structure on sequential spectra

The model category structure on sequential spectra which presents stable homotopy theory is the “stable model structure” discussed below. Its fibrant-cofibrant objects are (in particular) Omega-spectra, hence are the proper spectrum objects among the pre-spectrum objects.

But for technical purposes it is useful to also be able to speak of a model structure on pre-spectra, which sees their homotopy theory as sequences of simplicial sets equipped with suspension maps, but not their stable structure. This is called the “strict model structure” for sequential spectra. It’s main point is that the stable model structure of interest arises fromit via left Bousfield localization.

Definition

Say that a homomorphism f :X Y f_\bullet \colon X_\bullet \to Y_\bullet in the category SeqSpec(sSet)SeqSpec(sSet), def. is

Proposition

The classes of morphisms in def. give the structure of a model category SeqSpec(sSet) strictSeqSpec(sSet)_{strict}, called the strict model structure on sequential spectra.

Moreover, this is

(Bousfield-Friedlander 78, prop. 2.2).

Proof

The representation of sequential spectra as diagram spectra says that the category of sequential spectra is equivalently an enriched functor category

SeqSpec(sSet)[StdSpheres,sSet */] SeqSpec(sSet) \simeq [StdSpheres, sSet^{\ast/}]

(this proposition). Accordingly, this carries the projective model structure on enriched functors, and unwinding the definitions, this gives the statement for the fibrations and the weak equivalences.

It only remains to check that the cofibrations are as claimed. To that end, consider a commuting square of sequential spectra

X h A f Y B . \array{ X_\bullet &\stackrel{h_\bullet}{\longrightarrow}& A_\bullet \\ \downarrow^{\mathrlap{f_\bullet}} && \downarrow \\ Y_\bullet &\longrightarrow& B_\bullet } \,.

By definition, this is equivalently a \mathbb{N}-collection of commuting diagrams of simplicial sets of the form

X n h n A n f n Y n B n \array{ X_n &\stackrel{h_n}{\longrightarrow}& A_n \\ \downarrow^{\mathrlap{f_n}} && \downarrow \\ Y_n &\longrightarrow& B_n }

such that all structure maps are respected.

X n σ n X X n+1 f n f n+1 Y n σ n Y Y n+1 B n σ n B B n+1X n σ n X X n+1 h n h n+1 A n σ n A A n+1 B n σ n B B n+1. \array{ X_n &\stackrel{\sigma_n^X}{\longrightarrow}& X_{n+1} \\ \downarrow^{\mathrlap{f_n}} && \downarrow^{\mathrlap{f_{n+1}}} \\ Y_n &\stackrel{\sigma_n^Y}{\longrightarrow}& Y_{n+1} \\ & \searrow && \searrow \\ && B_n &\stackrel{\sigma_n^B}{\longrightarrow}& B_{n+1} } \;\;\; \Rightarrow \;\;\; \array{ X_n &\stackrel{\sigma_n^X}{\longrightarrow}& X_{n+1} \\ & \searrow^{\mathrlap{h_n}} && \searrow^{\mathrlap{h_{n+1}}} \\ && A_n &\stackrel{\sigma_n^A}{\longrightarrow}& A_{n+1} \\ && \downarrow && \downarrow \\ && B_n &\stackrel{\sigma_n^B}{\longrightarrow}& B_{n+1} } \,.

Hence a lifting in the original diagram is a lifting in each degree nn, such that the lifting in degree n+1n+1 makes these diagrams of structure maps commute.

Since components are parameterized over \mathbb{N}, this condition has solutions by induction. First of all there must be an ordinary lifting in degree 0. Then assume a lifting l nl_n in degree nn has been found

X n h n A n f n l n Y n B n \array{ X_n &\stackrel{h_n}{\longrightarrow}& A_n \\ \downarrow^{\mathrlap{f_n}} &\nearrow_{\mathrlap{l_n}}& \downarrow \\ Y_n &\longrightarrow& B_n }

the lifting l n+1l_{n+1} in the next degree has to also make the following diagram commute

X n σ n X X n+1 f n h n+1 Y n σ n Y Y n+1 l n l n+1 A n σ n A A n+1. \array{ X_n &\stackrel{\sigma_n^X}{\longrightarrow}& X_{n+1} \\ \downarrow^{\mathrlap{f_n}} && \downarrow^{\mathrlap{h_{n+1}}} & \searrow \\ Y_n &\stackrel{\sigma_n^Y}{\longrightarrow}& Y_{n+1} && \\ & \searrow^{\mathrlap{l_n}} && \searrow^{\mathrlap{l_{n+1}}} & \downarrow \\ && A_n &\stackrel{\sigma_n^A}{\longrightarrow}& A_{n+1} } \,.

This is a cocone under the the commuting square for the structure maps, and therefore the outer diagram is equivalently a morphism out of the domain of the pushout product f nσ n Xf_n \Box \sigma_n^X, while the compatible lift l n+1l_{n+1} is equivalently a lift against this pushout product:

Y nX nX n+1 (σ n Al n,h n+1) A n+1 l n+1 Y n+1 B n+1. \array{ Y_n \underset{X_n}{\sqcup} X_{n+1} &\stackrel{(\sigma_n^A l_n,h_{n+1})}{\longrightarrow}& A_{n+1} \\ \downarrow &{}^{\mathllap{l_{n+1}}}\nearrow& \downarrow \\ Y_{n+1} &\stackrel{}{\longrightarrow}& B_{n+1} } \,.

The stable model structure on sequential spectra

Definition

Say that a homomorphism f :X Y f_\bullet \colon X_\bullet \to Y_\bullet in the category SeqSpec(sSet)SeqSpec(sSet), def. is

Proposition

The classes of morphisms in def. give the structure of a model category SeqSpec(sSet) stableSeqSpec(sSet)_{stable}, called the stable model structure on sequential spectra.

Moreover, this is

(Bousfield-Friedlander 78, theorem 2.3).

Proof

By corollary , the stable model structure SeqSpectra(sSet) stableSeqSpectra(sSet)_{stable} is, if indeed it exists, the left Bousfield localization of the strict model structure of prop. at the morphisms that become weak equivalences under the spectrification functor Q:SeqSpectra(sSet)SeqSpectra(sSet)Q \colon SeqSpectra(sSet) \longrightarrow SeqSpectra(sSet), def. . By prop. QQ satisfies the conditions of the Bousfield-Friedlander theorem, and this implies the claim.

Remark

A spectrum XSeqSpec(sSet) stableX \in SeqSpec(sSet)_{stable} is

  • fibrant precisely if it is an Omega-spectrum, def. , and each X nX_n is a Kan complex;

  • cofibrant precisely if all the structure maps S 1X nX n+1S^1 \wedge X_n \to X_{n+1} are cofibrations of simplicial sets, i.e. monomorphisms.

Properties

Fibrations and cofibrations

Proposition

A sequential spectrum XSeqSpec(sSet) stableX\in SeqSpec(sSet)_{stable} is cofibrant precisely if all its structure morphisms S 1X nX n+1S^1 \wedge X_n \to X_{n+1} are monomorphisms.

Proof

A morphism *X\ast \to X is a cofibration according to def. (in either the strict or stable model structure, they have the same cofibrations) if

  1. X 0X_0 is cofibrant; this is no condition in sSet;

  2. * n+1S 1* nS 1X nX n+1 \ast_{n+1}\underset{S^1 \wedge \ast_n}{\coprod} S^1 \wedge X_n \longrightarrow X_{n+1}

    is a cofibration. But in this case the pushout reduces to just its second summand, and so this is now equivalent to

    S 1X nX n+1 S^1 \wedge X_n \longrightarrow X_{n+1}

    being cofibrations; hence inclusions.

Relation to sequential spectra in TopTop and to combinatorial spectra

Proposition

There is a zig-zag of Quillen equivalences relating the Bousfield-Friedlander model structure SeqSpec(sSet) stableSeqSpec(sSet)_{stable}, def. , prop. with standard model structures on sequential spectra in topological spaces (the model structure on topological sequential spectra) and with Kan’s combinatorial spectra.

(Bousfield-Friedlander 78, section 2.5).

Relation to symmetric spectra

There is a Quillen equivalence to the model structure on symmetric spectra (Hovey-Shipley-Smith 00, section 4.3, Mandell-May-Schwede-Shipley 01, theorem 0.1).

Relation to excisive functors

There is a Quillen equivalence between the Bousfield-Friedlander model structure and a model structure for excisive functors (Lydakis 98).

Definition

Write

Write

sSet */u() +sSet sSet^{\ast/} \stackrel{\overset{(-)_+}{\longleftarrow}}{\underset{u}{\longrightarrow}} sSet

for the free-forgetful adjunction, where the left adjoint functor () +(-)_+ freely adjoins a base point.

Write

:sSet */×sSet */sSet */ \wedge \colon sSet^{\ast/} \times sSet^{\ast/} \longrightarrow sSet^{\ast/}

for the smash product of pointed simplicial sets, similarly for its restriction to sSet fin *sSet_{fin}^{\ast}:

XYcofib(((u(X),*)(*,u(Y)))u(X)×u(Y)). X \wedge Y \coloneqq cofib\left( \; \left(\, (u(X),\ast) \sqcup (\ast, u(Y)) \,\right) \longrightarrow u(X) \times u(Y) \; \right) \,.

This gives sSet */sSet^{\ast/} and sSet fin */sSet^{\ast/}_{fin} the structure of a closed monoidal category and we write

[,] *:(sSet */) op×sSet */sSet */ [-,-]_\ast \;\colon\; (sSet^{\ast/})^{op} \times sSet^{\ast/} \longrightarrow sSet^{\ast/}

for the corresponding internal hom, the pointed function complex functor.

We regard all the categories in def. canonically as simplicially enriched categories, and in fact regard sSet */sSet^{\ast/} and sSet fin */sSet^{\ast/}_{fin} as sSet */sSet^{\ast/}-enriched categories.

The category that supports a model structure for excisive functors is the sSet */sSet^{\ast/}-enriched functor category

[sSet fin */,sSet */]. [sSet^{\ast/}_{fin}, sSet^{\ast/}] \,.

(Lydakis 98, example 3.8, def. 4.4)

In order to compare this to to sequential spectra consider also the following variant.

Definition

Write S std 1Δ[1]/Δ[1]sSet */S^1_{std} \coloneqq \Delta[1]/\partial\Delta[1]\in sSet^{\ast/} for the standard minimal pointed simplicial 1-sphere.

Write

ι:StdSpheressSet fin */ \iota \;\colon\; StdSpheres \longrightarrow sSet^{\ast/}_{fin}

for the non-full sSet */sSet^{\ast/}-enriched subcategory of pointed simplicial finite sets, def. whose

  • objects are the smash product powers S std n(S std 1) nS^n_{std} \coloneqq (S^1_{std})^{\wedge^n} (the standard minimal simplicial n-spheres);

  • hom-objects are

    [S std n,S std n+k] StdSpheres{* for k<0 im(S std k[S std n,S std n+k] sSet fin */) otherwise [S^{n}_{std}, S^{n+k}_{std}]_{StdSpheres} \coloneqq \left\{ \array{ \ast & for & k \lt 0 \\ im(S^{k}_{std} \stackrel{}{\to} [S^n_{std}, S^{n+k}_{std}]_{sSet^{\ast/}_{fin}}) & otherwise } \right.

(Lydakis 98, def. 4.2)

Proposition

There is an sSet */sSet^{\ast/}-enriched functor

() seq:[StdSpheres,sSet */]SeqSpec(sSet) (-)^seq \;\colon\; [StdSpheres,sSet^{\ast/}] \longrightarrow SeqSpec(sSet)

(from the category of sSet */sSet^{\ast/}-enriched copresheaves on the categories of standard simplicial spheres of def. to the category of sequential spectra in sSet, def. ) given on objects by sending X[StdSpheres,sSet */]X \in [StdSpheres,sSet^{\ast/}] to the sequential spectrum X seqX^{seq} with components

X n seqX(S std n) X^{seq}_n \coloneqq X(S^n_{std})

and with structure maps

S std 1X n seqσ nX n seqS std 1[X n seq,X n+1 seq] \frac{S^1_{std} \wedge X^{seq}_n \stackrel{\sigma_n}{\longrightarrow} X^{seq}_n}{S^1_{std} \longrightarrow [X^{seq}_n, X^{seq}_{n+1}]}

given by

S std 1id˜[S std n,S std n+1]X S std n,S std n+1[X n seq,X n+1 seq]. S^1_{std} \stackrel{\widetilde{id}}{\longrightarrow} [S^n_{std}, S^{n+1}_{std}] \stackrel{X_{S^n_{std}, S^{n+1}_{std}}}{\longrightarrow} [X^{seq}_n, X^{seq}_{n+1}] \,.

This is an sSet */sSet^{\ast/} enriched equivalence of categories.

(Lydakis 98, prop. 4.3)

Proposition

The adjunction

(ι *ι *):[sSet fin */,sSet */] Lyι *ι *[StdSpheres,sSet */]() seqSeqSpec(sSet) stable (\iota_\ast \dashv \iota^\ast) \;\colon\; [sSet^{\ast/}_{fin}, sSet^{\ast/}]_{Ly} \stackrel{\overset{\iota_\ast}{\longleftarrow}}{\underset{\iota^\ast}{\longrightarrow}} [StdSpheres, sSet^{\ast/}] \underoverset{\simeq}{(-)^{seq}}{\longrightarrow} SeqSpec(sSet)_{stable}

(given by restriction ι *\iota^\ast along the defining inclusion ι\iota of def. and by left Kan extension ι *\iota_\ast along ι\iota and combined with the equivalence () seq(-)^{seq} of prop. ) is a Quillen adjunction and in fact a Quillen equivalence between the Bousfield-Friedlander model structure on sequential spectra and Lydakis’ model structure for excisive functors.

(Lydakis 98, theorem 11.3) For more details see at model structure for excisive functors.

References

The original construction is due to

  • Aldridge Bousfield, Eric Friedlander, Homotopy theory of Γ\Gamma-spaces, spectra, and bisimplicial sets, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. (pdf)

Generalization of this model structure from sequential pre-spectra in sSet*/^{\ast/} to sequential spectra in more general proper pointed simplicial model categories is in

  • Stefan Schwede, Spectra in model categories and applications to the algebraic cotangent complex, Journal of Pure and Applied Algebra 120 (1997) 77-104 (pdf)

Discussion of the Quillen equivalence to the model structure on excisive functors (which does have a symmetric smash product of spectra) is in

  • Lydakis, Simplicial functors and stable homotopy theory Preprint, available via Hopf archive, 1998 (pdf)

Discussion of the Quillen equivalence to the model structure on symmetric spectra is in

Last revised on April 15, 2016 at 10:41:52. See the history of this page for a list of all contributions to it.