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External sequential simplicial Omega replacement

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

Statement

Let K=Δ¹/∂Δ¹ and let X be a sequential pointed simplicial prespectrum, meaning pointed simplicial sets $X_{n}$ and maps K∧ $X_{n}$ → $X_{n + 1}$ , with each $X_{n}$ a Kan complex. A Kan complex has fillers for all simplicial horns. Write ΩZ=Map_*(K,Z), let s denote the level tail, and put $R = s Ω$ . The adjoints of the structure maps form a natural map ι:X→RX. Define QX=colim_m $R^{m} X$ using the maps $R^{m} ι$ . Then QX is level Kan, its adjoint structure maps are weak equivalences, X→QX is a stable equivalence, and (QX)n=colim_m Ω^mX{n+m}. A stable equivalence here is a weak equivalence in Hovey's sequential stable model structure.

Source contract

Given: K and a level-Kan sequential pointed simplicial prespectrum X.

Goal: Import the functorial Omega replacement and its precise sequential formula.

[F1] Hovey's Definition 4.2, Lemma 4.3, Proposition 4.4 and Corollary 4.8 give the construction and conclusions under the cited hypotheses.

External input

Import [F1] specialized to pointed simplicial sets and the adjunction K∧(−) ⊣ Map_*(K,−). The finite simplicial circle makes Ω preserve filtered colimits, and filtered colimits of sets commute with finite limits. Hovey's Lemma 4.3 identifies ι_{RX} with Rι_X. The formula is not applied directly to topological spaces, and no symmetric-spectrum or model-category equivalence is asserted.

Depends on

Mark Hovey, Spectra and symmetric spectra in general model categories · §4, Definition 4.2, Lemma 4.3, Proposition 4.4, Corollary 4.8 and Theorem 4.9, PDF pp. 17–20

Used by · 1

Omega replacement preserves sequential stable groups

Dependency tree · 1 level

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.

External dependencies

Mark Hovey, Spectra and symmetric spectra in general model categories · §4, Definition 4.2, Lemma 4.3, Proposition 4.4, Corollary 4.8 and Theorem 4.9, PDF pp. 17–20

For level-fibrant sequential spectra in pointed simplicial sets, R=sU and Q=colim R^m give a level-fibrant Omega-spectrum; X→QX is a stable equivalence, and (QX)_n=colim_m Ω^m X_{n+m}.

Hypotheses: The general result assumes an almost finitely generated model category, sequential colimits preserving finite products, and a right adjoint U preserving sequential colimits. In pointed simplicial sets these conditions hold for U=Map_*(Δ¹/∂Δ¹,−); level fibrancy means each level is Kan.

Reading limits: The current author inspected Definition 4.2, Lemma 4.3, Proposition 4.4, Corollary 4.8 and Theorem 4.9, including the proof of Lemma 4.3 and the displayed proposition statement. The supplied authoring assistance reports a broader reading of §§3–4. Model-category foundations and verification of all general hypotheses are not reconstructed here.

Sources

Mark Hovey, Spectra and symmetric spectra in general model categories · §4, Definition 4.2, Lemma 4.3, Proposition 4.4, Corollary 4.8 and Theorem 4.9, PDF pp. 17–20