Pointed simplicial constructions and sequential model structures

PDF p. 3: Filtered S1 spectra: stable enhancement prerequisites

Assume ZFC. Let ${\mathrm{s}\mathrm{S}\mathrm{e}\mathrm{t}}_{\ast }$ be the category of simplicial sets with a specified map $\ast =\mathrm{\Delta }[0]\to A$ and maps preserving it. Put $T_{+}=T⨿\ast$ for an unpointed simplicial set $T$. For pointed $A,B$, define $A\wedge B=(A\times B)\mathrm{/}(A\times \ast \cup \ast \times B)$, degreewise, and define ${\mathrm{Map}}_{\ast }(A,B{)}_q={\mathrm{Hom}}_{{\mathrm{s}\mathrm{S}\mathrm{e}\mathrm{t}}_{\ast }}(A\wedge \mathrm{\Delta }[q{]}_{+},B)$, with simplicial operators induced by those of $\mathrm{\Delta }[q]$ and basepoint the constant map. These constructions give the smash–mapping adjunction. Set $K=\mathrm{\Delta }[1]\mathrm{/}\mathrm{\partial }\mathrm{\Delta }[1]$, based at the collapsed boundary, and $\mathrm{\Omega }B={\mathrm{Map}}_{\ast }(K,B)$.

For clarity, $\mathrm{\mid }T\mathrm{\mid }$ means the quotient of ${\coprod }_{q\ge 0}{T}_q\times \{(t_0,\dots ,t_q)\in {\mathbb{R}}_{\ge 0}^{q+1}:\sum t_i=1\}$ by $(T(\alpha )x,t)\sim (x,{\alpha }_{\ast }t)$ for each order-preserving $\alpha :[p]\to [q]$, where ${\alpha }_{\ast }$ sums coordinates over each fiber. The Kan model structure on ${\mathrm{s}\mathrm{S}\mathrm{e}\mathrm{t}}_{\ast }$ has weak equivalences the maps whose realizations are weak homotopy equivalences, fibrations the underlying Kan fibrations, and cofibrations the degreewise injections. It has functorial model factorizations; all objects are cofibrant and its fibrant objects are the Kan complexes. Generating cofibrations are $I_{+}=\{\mathrm{\partial }\mathrm{\Delta }[q{]}_{+}\to \mathrm{\Delta }[q{]}_{+}:q\ge 0\}$ and generating trivial cofibrations are $J_{+}=\{{\mathrm{\Lambda }}^r[q{]}_{+}\to \mathrm{\Delta }[q{]}_{+}:q\ge 1,0\le r\le q\}$. Here generation means that trivial fibrations and fibrations are detected by the right lifting properties against $I_{+}$ and $J_{+}$, respectively, with the corresponding small-object factorizations. All generator domains and codomains have finitely many nondegenerate simplices. Smash product satisfies the pushout-product axiom: for cofibrations $i:A\to B$ and $j:C\to D$, $(B\wedge C){⨿}_{A\wedge C}(A\wedge D)\to B\wedge D$ is a cofibration, trivial if either $i$ or $j$ is trivial. In particular $K\wedge (-)⊣\mathrm{\Omega }$ is a Quillen adjunction.

A sequential prespectrum is a sequence $X_n$ in ${\mathrm{s}\mathrm{S}\mathrm{e}\mathrm{t}}_{\ast }$, $n\ge 0$, with sphere-first maps $K\wedge X_n\to X_{n+1}$; a map of prespectra is a sequence commuting with these maps. This category has the projective model structure, whose weak equivalences and fibrations are levelwise and whose cofibrations have the left lifting property against all levelwise trivial fibrations. It also has the sequential stable model structure, with the same cofibrations, and fibrant objects precisely the level-Kan prespectra whose adjoint maps $X_n\to \mathrm{\Omega }{X}_{n+1}$ are weak equivalences. To specify its weak equivalences concretely, let $\mathcal{M}(A,E{)}_q$ be the set of prespectrum maps $A\wedge \mathrm{\Delta }[q{]}_{+}\to E$, where $(A\wedge T{)}_n=A_n\wedge T$ and its structure map is $K\wedge (A_n\wedge T)\cong (K\wedge A_n)\wedge T\to A_{n+1}\wedge T$. For a functorial projective cofibrant replacement $PA\to A$, a map $f:A\to B$ is a stable equivalence exactly when $\mathcal{M}(PB,E)\to \mathcal{M}(PA,E)$ is a weak equivalence of simplicial sets for every such fibrant $E$. Stable fibrations are the maps with the right lifting property against the cofibrations that are stable equivalences.

Simplicial sets, homotopies and trivial Kan fibrations

Simplicial horns and Kan fibrations

Model categories and Quillen adjunctions

Weak homotopy equivalence

The Axiom of Choice

The Kan model structure, its pointed generators, the smash–mapping adjunction and its pushout-product property are imported from Hovey's Model categories at the locators in reference 0. The sequential projective and stable structures and the mapping-space criterion are imported from reference 1, specifically its pointed-simplicial specialization. Its Theorem 5.7 supplies the levelwise tensor and compatible-map enrichment used in $\mathcal{M}$; the projective cofibrant and level-fibrant inputs make this the derived mapping space of §2. The displayed quotient and mapping formulas fix the interfaces of those imports. No assertion about symmetric spectra or motivic localizations is included.

The Axiom of Choice is allowed in the imported small-object factorizations and model-structure existence constructions, including simultaneous selections of lifts. No independent reconstruction of those substantial existence arguments is claimed. In particular the stable existence premise is the specific simplicial case of Corollary 3.5, rather than an unverified invocation of the general left proper cellular localization theorem.