nLab Bousfield-Friedlander model structure

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

$$\wedge \;\colon\; sSet^{\ast/} \times sSet^{\ast/} \longrightarrow sSet^{\ast/}$$

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 $X_\bullet$

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

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

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

commute.

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

Example

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

$$(X \wedge K_+)_n \coloneqq (X_n \wedge K_+)$$

and with structure maps given by

$$S^1 \wedge (X_n \wedge K_+) \simeq (S^1 \wedge X_n) \wedge K_+ \stackrel{\sigma_n \wedge K_+}{\longrightarrow} X_{n+1}\wedge K_+ \,.$$

Proposition

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

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

Definition

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

$$\pi_\bullet(X) \coloneqq \underset{\longrightarrow}{\lim}_k \pi_{\bullet+k}({\vert X_n \vert}) \,.$$

This constitutes a functor

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

Definition

A morphism $f \colon X \longrightarrow Y$ 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

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

Omega-spectra

Definition

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

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

of the structure maps ${\vert \sigma_n\vert}$ are weak homotopy equivalences.

Remark

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

$$\pi_k(X) \simeq \simeq \left\{ \array{ \pi_k {\vert X_0 \vert} & if\; k \geq 0 \\ \pi_0 {\vert X_k \vert} & if \; k \lt 0 } \right. \,.$$

Definition

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

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

where $Sing$ denotes the singular simplicial complex functor.

This constitutes an endofunctor

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

Write

$$\eta \;\colon\; id \longrightarrow Q$$

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

$$(\eta_X)_n \;\colon\; X_n \longrightarrow Sing{\vert X_n\vert} \stackrel{\iota_0}{\longrightarrow} \underset{\longrightarrow}{\lim}_{k } Sing \Omega^k {\vert X_{n+k}\vert } \,.$$

Proposition

The spectrification of def. satisfies

1. $Q X$ is an Omega-spectrum, def. ;

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

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

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

Corollary

A homomorphism of sequential spectra, def. , is a stable weak homotopy equivalence, def. , precisely if its spectrification $Q f$ , 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 $f_\bullet \colon X_\bullet \to Y_\bullet$ in the category $SeqSpec(sSet)$, def. is

• a strict weak equivalence if each component $f_n \colon X_n \to Y_n$ is a weak equivalence in the classical model structure on simplicial sets (hence a weak homotopy equivalence of geometric realizations);

• a strict fibration if each component $f_n \colon X_n \to Y_n$ is a fibration in the classical model structure on simplicial sets (hence a Kan fibration);

• a strict cofibration if the simplicial maps $f_0\colon X_0 \to Y_0$ as well as all pushout products of $f_n$ with the structure maps of $X$

  $$X_{n+1}\underset{S^1 \wedge X_n}{\coprod} S^1 \wedge Y_n \longrightarrow Y_{n+1}$$

  are cofibrations of simplicial sets in the classical model structure on simplicial sets (i.e.: monomorphisms of simplicial sets);

Proposition

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

Moreover, this is

• a proper model category;

• a simplicial model category with respect to the simplicial enrichment of prop. .

(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

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

(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

$$\array{ X_\bullet &\stackrel{h_\bullet}{\longrightarrow}& A_\bullet \\ \downarrow^{\mathrlap{f_\bullet}} && \downarrow \\ Y_\bullet &\longrightarrow& B_\bullet } \,.$$

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

$$\array{ X_n &\stackrel{h_n}{\longrightarrow}& A_n \\ \downarrow^{\mathrlap{f_n}} && \downarrow \\ Y_n &\longrightarrow& B_n }$$

such that all structure maps are respected.

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

Hence a lifting in the original diagram is a lifting in each degree $n$, such that the lifting in degree $n+1$ 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 $l_n$ in degree $n$ has been found

$$\array{ X_n &\stackrel{h_n}{\longrightarrow}& A_n \\ \downarrow^{\mathrlap{f_n}} &\nearrow_{\mathrlap{l_n}}& \downarrow \\ Y_n &\longrightarrow& B_n }$$

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

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

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 $f_n \Box \sigma_n^X$, while the compatible lift $l_{n+1}$ is equivalently a lift against this pushout product:

$$\array{ Y_n \underset{X_n}{\sqcup} X_{n+1} &\stackrel{(\sigma_n^A l_n,h_{n+1})}{\longrightarrow}& A_{n+1} \\ \downarrow &{}^{\mathllap{l_{n+1}}}\nearrow& \downarrow \\ Y_{n+1} &\stackrel{}{\longrightarrow}& B_{n+1} } \,.$$

The stable model structure on sequential spectra

Definition

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

• a stable weak equivalence if it is a stable weak homotopy equivalence, def. ;

• a stable cofibration if the simplicial maps $f_0\colon X_0 \to Y_0$ as well as all pushout products of $f_n$ with the structure maps of $X$

  $$X_{n+1}\underset{S^1 \wedge X_n}{\coprod} S^1 \wedge Y_n \longrightarrow Y_{n+1}$$

  are cofibrations of simplicial sets in the classical model structure on simplicial sets (i.e.: monomorphisms of simplicial sets);

• a stable fibration if it is degreewise a fibration of simplicial sets, hence degreewise a Kan fibration, and if in addition the naturality squares of the spectrification, def. ,

  $$\array{ X_n &\stackrel{}{\longrightarrow}& (Q X)_n \\ \downarrow^{\mathrlap{f_n}} && \downarrow^{\mathrlap{Q f_n}} \\ Y_n &\stackrel{}{\longrightarrow}& (Q Y)_n }$$

  are homotopy pullback squares (with respect to the classical model structure on simplicial sets).

Proposition

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

Moreover, this is

• a proper model category;

• a simplicial model category with respect to the simplicial enrichment of prop. .

(Bousfield-Friedlander 78, theorem 2.3).

Proof

By corollary , the stable model structure $SeqSpectra(sSet)_{stable}$ 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 $Q \colon SeqSpectra(sSet) \longrightarrow SeqSpectra(sSet)$, def. . By prop. $Q$ satisfies the conditions of the Bousfield-Friedlander theorem, and this implies the claim.

Remark

A spectrum $X \in SeqSpec(sSet)_{stable}$ is

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

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

Properties

Fibrations and cofibrations

Proposition

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

Proof

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

1. $X_0$ is cofibrant; this is no condition in sSet;

2. $$\ast_{n+1}\underset{S^1 \wedge \ast_n}{\coprod} S^1 \wedge X_n \longrightarrow X_{n+1}$$

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

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

   being cofibrations; hence inclusions.

Relation to sequential spectra in $Top$ and to combinatorial spectra

Proposition

There is a zig-zag of Quillen equivalences relating the Bousfield-Friedlander model structure $SeqSpec(sSet)_{stable}$, 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

• sSet for the category of simplicial sets;

• $sSet^{\ast/}$ for the category of pointed simplicial sets;

• $sSet_{fin}^{\ast/}\simeq s(FinSet)^{\ast/} \hookrightarrow sSet^{\ast/}$ for the full subcategory of pointed simplicial finite sets.

Write

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

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

Write

$$\wedge \colon sSet^{\ast/} \times sSet^{\ast/} \longrightarrow sSet^{\ast/}$$

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

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

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

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

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 $sSet^{\ast/}$ and $sSet^{\ast/}_{fin}$ as $sSet^{\ast/}$-enriched categories.

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

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

(Lydakis 98, example 3.8, def. 4.4)

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

Definition

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

Write

$$\iota \;\colon\; StdSpheres \longrightarrow sSet^{\ast/}_{fin}$$

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

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

• hom-objects are

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

(Lydakis 98, def. 4.2)

Proposition

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

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

(from the category of $sSet^{\ast/}$-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 $X \in [StdSpheres,sSet^{\ast/}]$ to the sequential spectrum $X^{seq}$ with components

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

and with structure maps

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

given by

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

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

(Lydakis 98, prop. 4.3)

Proposition

The adjunction

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

(given by restriction $\iota^\ast$ along the defining inclusion $\iota$ of def. and by left Kan extension $\iota_\ast$ along $\iota$ and combined with the equivalence $(-)^{seq}$ 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.

Related concepts

• model structure on topological sequential spectra

References

The original construction is due to

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

• Mark Hovey, Brooke Shipley, Jeff Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), 149-208 (arXiv:math/9801077)

• Michael Mandell, Peter May, Stefan Schwede, Brooke Shipley, theorem 0.1 of Model categories of diagram spectra, Proceedings of the London Mathematical Society, 82 (2001), 441-512 (pdf)